advanced physics course chapter 6: circular motion€¦ · advanced physics course chapter 6:...

A D V A N C E D P H Y S I C S C O U R S E

C H A P T E R 6 :

C I R C U L A R M O T I O N

FOR HIGH SCHOOL PHYSICS CURRICULUM AND ALSO THE PREPARATION OF ACT, DSST, AND AP EXAMS

This is a complete video-based high school physics course that includes videos, labs, and hands-on learning.

You can use it as your core high school physics curriculum, or as a college-level test prep course. Either way,

you’ll find that this course will not only guide you through every step preparing for college and advanced

placement exams in the field of physics, but also give you in hands-on lab practice so you have a full and

complete education in physics. Includes text reading, exercises, lab worksheets, homework and answer keys.

BY AURORA LIPPER ∙ SUPERCHARGED SCIENCE 2017

© 2017 Supercharged Science

TABLE OF CONTENTS

Material List ............................................................................................................................................................................................................ 4

Introduction ............................................................................................................................................................................................................ 5

Characteristics for Circular Motion............................................................................................................................................................... 6

Introduction to Circular Motion ..................................................................................................................................................................... 7

Calculating Average Speed ............................................................................................................................................................................... 8

.Acceleration in a Circle ..................................................................................................................................................................................... 8

Acceleration in a Circle ...................................................................................................................................................................................... 9

Accelerometer ..................................................................................................................................................................................................... 10

Rotating Candles ................................................................................................................................................................................................ 18

Centripetal Force ............................................................................................................................................................................................... 19

Earth-Satellite System ..................................................................................................................................................................................... 20

Another Earth-Satellite System Example ................................................................................................................................................ 21

Centrifugal Force ............................................................................................................................................................................................... 22

Circular Motion with a Car ............................................................................................................................................................................ 24

Circular Motion with a Kid ............................................................................................................................................................................ 25

Favorite Amusement Park Ride................................................................................................................................................................... 26

Circular Motion and Friction ........................................................................................................................................................................ 27

Swinging Bucket ................................................................................................................................................................................................ 28

Clothoid Loops .................................................................................................................................................................................................... 29

Roller Coaster Maneuvers.............................................................................................................................................................................. 30

Roller Coaster Activity .................................................................................................................................................................................... 31

Speed Skating ...................................................................................................................................................................................................... 36

Unbanked Car Turn without Friction ........................................................................................................................................................ 37

Banked Car Turn with Friction .................................................................................................................................................................... 38

Universal Gravitation ....................................................................................................................................................................................... 39

Inverse Square Law .......................................................................................................................................................................................... 41

Applying the Inverse Square Law ............................................................................................................................................................... 42

Cavendish Experiment .................................................................................................................................................................................... 43

Star Wobble ......................................................................................................................................................................................................... 44

If the Earth Gained Weight, Would You? ................................................................................................................................................. 50

Turning the Sun into a Black Hole .............................................................................................................................................................. 51

Gravity and Inertia ............................................................................................................................................................................................ 52

Rockets and Gravity ......................................................................................................................................................................................... 59

Gravity for Different Objects ......................................................................................................................................................................... 60


Real Rocket Launch .......................................................................................................................................................................................... 61

The Earth’s Crust is Not Uniform ................................................................................................................................................................ 62

The Earth is not a Sphere ............................................................................................................................................................................... 63

Planetary and Satellite Motion..................................................................................................................................................................... 64

Fun Activity with Kepler’s Laws .................................................................................................................................................................. 65

Elliptical Orbits ................................................................................................................................................................................................... 70

Applying Kepler’s Laws .................................................................................................................................................................................. 71

Jupiter’s Moons ................................................................................................................................................................................................... 72

New Planets ......................................................................................................................................................................................................... 73

Why Satellites Stay in Orbit ........................................................................................................................................................................... 74

Satellite Crash! .................................................................................................................................................................................................... 75

Saturn’s Moons ................................................................................................................................................................................................... 80

Orbits of Satellites ............................................................................................................................................................................................. 81

Orbital Mechanics .............................................................................................................................................................................................. 82

Equations for Circular Motion ...................................................................................................................................................................... 83

How Fast is the Moon? .................................................................................................................................................................................... 84

Callisto ................................................................................................................................................................................................................... 85

Space Station Speed .......................................................................................................................................................................................... 86

Halley’s Comet .................................................................................................................................................................................................... 87

Weightlessness ................................................................................................................................................................................................... 88

Physics Fun in an Elevator ............................................................................................................................................................................. 89

Bust a Myth .......................................................................................................................................................................................................... 90

Binary System ..................................................................................................................................................................................................... 91

What’s Up in the Sky? ...................................................................................................................................................................................... 97

Build a Solar System ...................................................................................................................................................................................... 105

Retrograde Motion ......................................................................................................................................................................................... 116

Potential Energy in the Stars ..................................................................................................................................................................... 122

Escape Speed .................................................................................................................................................................................................... 123

Conservation of Mechanical Energy ....................................................................................................................................................... 124

Geosynchronous Orbit.................................................................................................................................................................................. 125

Gravitational Potential Energy ................................................................................................................................................................. 126

Rocketship Races ............................................................................................................................................................................................ 127

Equivalence Principle ................................................................................................................................................................................... 128

Homeowrk Problems with Solutions ..................................................................................................................................................... 129


MATERIAL LIST

While you can do the entire course entirely on paper, it’s not really recommended since physics is based in real-world observations and experiments! Here’s the list of materials you need in order to complete all the experiments in this unit. Please note: you do not have to do ALL the experiments in the course to have an outstanding science education. Simply pick and choose the ones you have the interest, time and budget for.

bucket outdoor area

clear tubing (about 12-18″ long)

nylon or metal barbed union that fits inside the tubing

empty soda bottle

clean wine cork

string

marbles

masking tape

3/4″ pipe foam insulation (NOT neoprene and NOT the kind with built-in adhesive tape)


INTRODUCTION

Circular motion is a little different from straight-line motion in a few different ways. Objects that move

in circles are roller coasters in a loop, satellites in orbit, DVDs spinning in a player, kids on a merry go

round, solar systems rotating in the galaxy, making a left turn in your car, water through a coiled hose,

and so much more. For any object that goes in a circle (or you can approximate it to a circle), you’ll want

to use this approach when solving problems. You can feel the effect of circular motion if you’ve ever

been in a car that suddenly turns right or left. You feel a push to the opposite side, right? If you are going

fast enough and you take the turn hard enough, you can actually get slammed against the door. So my

question to you is: who pushed you? That’s what this lesson is all about.

Be sure to take out a notebook and copy down each example problem right along with me so you take

good notes as you go along. It’s a totally different experience when you are actively involved by writing

down and working through each problem rather than passively sitting back and watching


CHARACTERISTICS FOR CIRCULAR MOTION

Circular motion is a little different from straight-line motion in a few different ways. Objects that move

in circles are roller coasters in a loop, satellites in orbit, DVDs spinning in a player, kids on a merry go

round, solar systems rotating in the galaxy, making a left turn in your car, water through a coiled hose,

and so much more.


INTRODUCTION TO CIRCULAR MOTION

Imagine driving your car in a circle, like you would when take a clover-leaf type of freeway exit, or make

a right turn on a green light. Here’s how the forces play out during the motion:


CALCULATING AVERAGE SPEED

For any object that goes in a circle (or you can approximate it to a circle), you’ll want to use this

approach when solving problems. You can feel the effect of circular motion if you’ve ever been in a car

that suddenly turns right or left. You feel a push to the opposite side, right? If you are going fast enough

and you take the turn hard enough, you can actually get slammed against the door. So my question to

you is: who pushed you? Let’s find out!

.


ACCELERATION IN A CIRCLE

An object that moves in a circle with constant speed (like driving your car in a big circle at 30 mph) is called uniform circular motion. Although the speed is constant (30 mph), the velocity, which is a vector and made up of speed and direction, is not constant. The velocity vector has the same speed (magnitude), but the direction keeps changing as your car moves around the circle. The direction is an arrow that’s tangent to the circle as long as the car is moving on a circular path. This means that the tangent arrow is constantly changing and pointing in a new direction. It’s a common assumption that if the speed is constant, then there’s no acceleration… right? Nope! If the velocity is constant, then there’s no acceleration. But for circular motion, it’s speed, not velocity that is constant. Velocity is changing as the car turns a corner because the direction is changing, which also means that there is acceleration also! An accelerating object changes it’s velocity… it can be changing it’s speed, direction, or both. So objects moving in a circle are accelerating because they are always changing their direction.


ACCELEROMETER

You’ve got homework now! Make this accelerometer and take it with you next time you go in the car (make sure you’re not the one driving so you can focus on your experiment). All you need is a water bottle, something that floats (like a cork or a piece of foam, and some string. To use the accelerometer: invert the bottle and try to make the cork move about. Remember – it is measuring acceleration, which is the change in speed. It will only move when your speed changes. You can do this experiment in a car while doing your other vehicle experiment: Why Bother With Seatbelts?. The trouble with this accelerometer is that there are no measurements you can take – it’s purely visual. This next activity is more accurate at measuring the number of g-s you pull in a sharp turn (whether in a vehicle or in a roller coaster!) One more university-level gadget for demonstrating the fascinating world of physical dynamics. This quick homemade device roughly measures acceleration in “g’s”. We used it to measure the g-force on roller coasters at Six Flags Magic Mountain, and it worked just as well as the expensive ones you buy in scientific catalogs! Now, next time an adult drives around town, hold the tube in your hand so that the water line starts at the zero mark. When she pulls a turn, see how far it sloshes up and tell her how many g’s she pulled. Kids love to use this device (not in a car) and have contests to see who could pull the most g’s while spinning in a circle super fast! Did you notice how these accelerometers measure (or indicate) the acceleration? With the cork accelerometer, the cork-water system travels together, but the cork has less mass and has less inertia, so it resists the acceleration the least and leans inside toward the center of the circle.


G-Force Overview: Have you ever been riding in a really fast car and you almost feel “pushed” back into your seat

becauseof how fast you start? Or been thrown forward when someone had to slam on the brakes? How about

pushed to the side when the car took a fast turn? So … who pushed you? That’s what this lab is all about.

What to Learn: You’ll learnabout centrifugal force, centripetal acceleration and g-force, and how to tell thedifference between them.

Materials

bucket

water

outdoor area

clear tubing (about 12-18″ long)

nylon or metal barbed union that fits inside the tubing

soda bottle (empty)

wine cork

string

Lab Time

1. To make the cork accelerometer, fill an empty soda bottle to the top with water.

2. Modify the soda bottle cap as follows: attach a string 8-10″ long to a clean cork, like the kind from a wine bottle.

3. Hot glue the free end of the string to the inside of the cap.

4. Place the cork and string inside the bottle and screw on the top (try to eliminate the air bubbles). The cork should be free to bob around when you hold the bottle upside-down.

5. Now race around and see if you can predict where the cork is going to go. Complete the data table.


G-Force Data Table 1

Activity You Did Which Way Do you Guess Cork Observations

the Cork Will Move?

Remember – it is measuring acceleration, which is the change in speed. It will only move when your speed changes. The trouble with this accelerometer is that there are no measurements you can take – it’s purely visual. This next activity in this lab is more accurate at measuring the number of g-s you pull in a sharp turn (whether in a vehicle or in a roller coaster!)

6. To make the g-force ring: (This quick homemade device roughly measures acceleration in “g’s.” We

used it to measure the g-force on roller coasters at Six Flags Magic Mountain, and it worked just as

well as the expensive ones you buy in scientific catalogs!)

7. Cut about a foot of tubing – the larger diameter tubing you can find, the easier it will be to read your measurements.


8. Fill your tube halfway with COLORED water (it’s impossible to read when it’s clear). Blue, green, red… your choice of food dye additive.

9. Make an O-shape using your barbed union to water-seal the junction.

10. Grab hold of one side and hold the circle vertical, with the barb-end pointing up.

11. Make sure there are equal amounts of water and air in your tube. Adjust if necessary.

12. Make a mark on the tube where the water meets the air with a black marker. This is your 0-g reading (relative, of course). No acceleration. Not a whole lot of fun.

13. Now, for your 1-g mark – measure up 45 degrees from the first mark. (If the top of the circle is 90

degrees, and the 0-g mark is zero degrees, find the halfway point and label it).

14. The 2-g mark is 22.5 degrees up from the 1-g mark.

15. 3-g mark is 11.25 degrees up from the last mark. And 4-g is 5.6 degrees up from the last mark. (See a pattern? You can prove this mathematically in college, and it’s kind of fun to figure out!)

16. Now, if you have access to a car with a driver or a playground with swings and spinning things, hold

the tube in your hand so that the water line starts at the zero mark. See how far it sloshes up when you

accelerate and read how many g’s you’ve pulled. We would have contests to see who could pull the

most g’s while spinning in a circle.

G-Force Data Table 2

Activity You Did How Many g’s Measured? Observations

Reading

G-force is really just the force that you experience by being accelerated. Have you ever been riding in

a really fast car and you almost feel “pushed” back into your seat because of how fast you start?


Probably not, because most of us don’t ride in cars that fast. More likely you’ve been “thrown”

forward when someone had to slam on the brakes, or you feel pushed to the side whenever someone

takes a fast turn. Whenever you feel these “pushes,” that means you are accelerating or changing

speeds.

The higher the acceleration, the harder the “push” feels. The reason I use quotation marks to

describe the push is to describe the “push” you feel when you experience the g-force. This push you

feel is actually just your own inertia wanting to maintain the motion (or rest) your body was

already in. When you suddenly stop, your body wants to continue moving forward, but your sense

of relative motion is set to the car as a stationary location.

When you feel yourself wanting to continue going forward, and the car is stopping you think of yourself as being “pushed” or “pulled” when you are really just trying to keep your original motion.

The same happens when you are turning, even when you are moving a constant speed. I know it

seems confusing, but we need remember that velocity is a vector, meaning that it takes into account

speed and direction. When you are turning at a constant speed, you are changing direction, and that

directional change means your velocity changes, so you have an acceleration.

The concept of the “push” or “pull” is the same as braking: Your body wants to continue to move in a straight line. When the car is turning to the left, your body wants to go straight, but in relative

motion to the car you appear to be moving to the right of the car, so you feel “pulled” to the right.

How does this all relate to “g-forces”? Well, like we learned before, the acceleration due to gravity is

9.8 m/s2, or 32 ft/s2. And the g-force that you experience is just a multiple of this number. For

example, if you experience an acceleration of 19.6 m/s2, you would divide this by 9.8 m/s2 to get 2

g’s. If you experience an acceleration of 48 ft/s2, you divide this by 32 ft/s2, and you get 1.5 g’s. So

it’s just how many multiples of gravity’s acceleration you experience.

How does acceleration relate to force? Through Newton’s Second Law: F = ma! But there are different kinds offorces (and thus acceleration): centripetal and centrifugal. How can you tell the difference?

Centripetal (translation = “center-seeking”) is the force needed to keep an object following a curved

path. Remember how objects will travel in a straight line unless they bump into something or have

another force acting on them (gravity, drag force, etc.)? Well, to keep the bucket of water swinging in

a curved arc, the centripetal force can be felt in the tension experienced by the handle (or your arm,

in our case). Swinging an object around on a string will cause the rope to undergo tension

(centripetal force), and if your rope isn’t strong enough, it will snap and break, sending the mass

flying off in a tangent (straight) line until gravity and drag force pull the object to a stop. This force is

proportional to the square of the speed - the faster you swing the object, the higher the force.

Centrifugal (translation = “center-fleeing”) force has two different definitions, which also causes

confusion. The inertial centrifugal force is the most widely referred to, and is purely mathematical,

having to do with calculating kinetic forces using reference frames, and is used with Newton’s laws

of motion. It’s often referred to as the ”fictitious force.”


Reactive centrifugal force happens when objects move in a curved path. This force is actually the

same magnitude as centripetal force, but in the opposite direction, and you can think of it as the

reaction force to the centripetal force. Think of how you stand on the Earth … your weight pushes

down on the Earth, and a reaction force (called the “normal” force) pushes up in reaction to your

weight, keeping you from falling to the center ofthe Earth. A centrifugal governor (spinning masses

that regulate the speed of an engine) and a centrifugal clutch (spinning disk with two masses

separated by a spring inside) are examples of this kind of force in action.

Imagine driving a car along a banked turn. The road exerts a centripetal force on the car, keeping the

car moving in a curved path (the “banked” turn). If you neglected to buckle your seat belt and the seats

have a fresh coat of Armor-All (making them slippery), then as the car turns along the banked curve,

you get “shoved” toward the door. But who pushed you? No one – your body wanted to continue in a

straight line but the car keeps moving in your path, turning your body in a curve. The push of your

weight on the door is the reactive centrifugal force, and the car pushing on you is the centripetal force.

What about the fictitious (inertial) centrifugal force? Well, if you imagine being inside the car as it is

banking with the windows blacked out, you suddenly feel a magical ”push” toward the door away from

the center of the bend. This “push” is the fictitious force invoked because the car’s motion and

acceleration is hidden from you (the observer) in the reference frame moving within the car.

Exercises Answer the questions below:

1. Which accelerometer was better at giving a visual representation of accelerating?

2. Which one do you prefer? Why?


3. What activity did you do that created the most acceleration?

4. What does that tell you about acceleration?


Answers to Exercises: G-Force

1. Which accelerometer was better at giving a visual representation of accelerating? (the liquid in the tube)

2. Which one do you prefer? Why? (The liquid in the tube actually gives a numerical measurement.)

3. What activity did you do that created the most acceleration? (spinning around in a circle)

4. What does that tell you about acceleration? (Centripetal acceleration is much higher and easier to achieve than linear acceleration.)


ROTATING CANDLES

Now this next experiment is a little dangerous (we’re going to be spinning flames in a circle), so I found a video by MIT that has a row of five candles sitting on a rotating platform (like a “lazy Susan”) so you can see how it works. The candles are placed inside a dome (or a glass jar) so that when we spin them, they aren’t affected by the moving air but purely by acceleration. So for this video above, a row of candles are inside a clear dome on a rotating platform. When the platform rotates, air inside the dome gets swung to the outer part of the dome, creating higher density air at the outer rim, and lower density air in the middle. The candle flames point inwards towards the middle because the hot gas in the flames always points towards lower density air. Source: http://video.mit.edu Now you’re beginning to understand how an object moving in a circle experiences acceleration, even if the speed is constant. So what direction is the acceleration vector? It’s pointed straight toward the center of that circle. Velocity is always tangent to the circle in the direction of the motion, and acceleration is always directed radially inward. Because of these two things, the acceleration that arises from traveling in a circle is called centripetal acceleration (a word created by Sir Isaac Newton). There’s no direct relationship between the acceleration and velocity vectors for a moving particle.


CENTRIPETAL FORCE

Do you remember when I asked you “Who pushed you?” when you were riding in a car that took a sharp turn? Well, the answer has to do with centripetal force. Centripetal (translation = “center-seeking” ) is the force needed to keep an object following a curved path. Remember how objects will travel in a straight line unless they bump into something or have another force acting on it like gravity, friction, or drag force? Imagine a car moving in a straight line at a constant speed. You’re inside the car, no seat belt, and the seat is slick enough for you to slide across easily. Now the car turns and drives again at constant speed but now on a circular path. When viewed from above the car, we see the car following a circle, and we see you wanting to keep moving in a straight line, but the car wall (door), moves into your path and exerts a force on you to keep you moving in a circle. The car door is pushing you into the circle. According to Newton’s second law of motion, if you are experiencing an acceleration you must also be experiencing a net force (F=ma). The direction of the net force is in the same direction as the acceleration, so for the example with you inside the car, there’s an inward force acting on you (from the car door) keeping you moving in a circle. If you have a bucket of water and you’re swinging it around your head, in order to keep a bucket of water swinging in a circle, the centripetal force can be felt in the tension experienced by the handle. Swinging an object around on a string will cause the rope to undergo tension (centripetal force), and if your rope isn’t strong enough, it will snap and break, sending the mass flying off in a tangential straight line until gravity and drag force pull the object to a stop. This force is proportional to the square of the speed, meaning that the faster you swing the object, the higher the magnitude of the force will be. Remember Newton’s First Law? The law of inertia? It states that objects in motion tend to stay in motion with the same speed and direction unless acted upon by an unbalanced/external force. Which means that objects naturally want to continue going their straight and merry way (like you did in a straight line when you were inside the car) until an unbalanced force causes it to turn speed up or stop. Can you see how an unbalanced force is required for objects to move in a circle? There has to be a force pushing on the object, keeping in on a circular path because otherwise, it’ll go off in a straight line!


EARTH-SATELLITE SYSTEM

Every object moving in a circle will experience a force pushing or pulling it toward the center of the circle. Whether it’s a car making a turn and the friction force from the road are acting on the wheels of the car, or a bucket is swung around your head and the tension of the rope keeps it moving in a circle, they all have to have a force keeping them moving in that circle, and that force is called centripetal force. Without it, objects could never change their direction. Because centripetal force is tangent to the velocity vector, the force can change the direction of an object without changing the magnitude.


ANOTHER EARTH-SATELLITE SYSTEM EXAMPLE

There’s a lot of confusion around the difference between centripetal and centrifugal force. The confusion usually starts with a thought like this: “When I go on a ride, I am getting thrown to the outside (if it’s like a fast merry-go-round) or being squashed down in my seat (if it’s like a roller coaster loop), but either way I am being pushed away from the center of the circle.” Can you imagine thinking that? Lots of people do! Now let’s see if we can punch a few holes in that thought so you can really see how it’s not true at all. First of all, without the inward force pushing on you to keep you in a circle, you would be going off in a straight line and not around the loop of the roller coaster. The track is exerting a force on you, pushing you toward the center of the circle. Now here’s a question for you: just because you feel like you’re being thrown, does that mean there has to be a force causing this? Is there any other way to explain that sensation? (Think Newton’s Laws!) Imagine again yourself in a car making a turn. If we had a video camera above the car, you’d see you wanting to continue in a straight line, but the car is now moving into your path and exerting a force on you, pushing you into a circle. That’s when you hit the door. The trick to really seeing this is to get out of yourself and into a different perspective! Einstein made a famous observation that described if you were in a rocket (without windows so you can’t view the outside world), you would not be able to tell if the rocket was in space accelerating, or if it were standing still and you were experiencing the same amount of acceleration due to gravity on a planet. Because of F=ma, you can experience these two situations and still feel the same and not be able to tell which is which!


CENTRIFUGAL FORCE

Centrifugal (translation = “center-fleeing”) force has two different definitions, which causes even more confusion. The inertial centrifugal force is the most widely referred to, and is purely mathematical, having to do with calculating kinetic forces using reference frames, and is used with Newton’s laws of motion. It’s often referred to as the ‘fictitious force’. Reactive centrifugal force happens when objects move in a curved path. This force is actually the same magnitude as centripetal force, but in the opposite direction, and you can think of it as the reaction force to the centripetal force. Think of how you stand on the Earth. Your weight pushes down on the Earth, and a reaction force (the “normal” force) pushes up in reaction to your weight, keeping you from falling to the center of the Earth. A centrifugal governor (spinning masses that regulate the speed of an engine) and a centrifugal clutch (spinning disk with two masses separated by a spring inside) are examples of this kind of force in action. Here’s anther example: imagine driving a car along a banked turn. The road exerts a centripetal force on the car, keeping the car moving in a curved path (the “banked” turn). If you neglected to buckle your seat belt and the seats have a fresh coat of Armor-All (making them slippery), then as the car turns along the banked curve, you get “shoved” toward the door. But who pushed you? No one – your body wanted to continue in a straight line but the car keeps moving in your path, turning your body in a curve. The push of your weight on the door is the reactive centrifugal force, and the car pushing on you is the centripetal force. What about the fictitious (inertial) centrifugal force? Well, if you imagine being inside the car as it is banking with the windows blacked out, you suddenly feel a magical ‘push’ toward the door away from the center of the bend. This “push” is the fictitious force invoked because the car’s motion and acceleration is hidden from you (the observer) in the reference frame moving within the car. James Watt invented a “centrifugal governor”, which is a closed loop mechanical device you’ll find in lawn mowers, cruise controls, and airplane propellers to automatically control the speed of these things. The heavy brass balls spin around, and the faster they go, the more they rise up, which increases the rotational energy of this device, and since it’s connected to the throttle of something like a lawn mower engine, it can be carefully set to maintain the same speed or output power of the engine. It’s an automatic feedback system that is purely mechanical. Source: MirkoJunge, Science Museum London. For circular motion, there are a couple of equations we will need to tackle physics problems that involve speed (v), acceleration (a), and force (F):


Here’s the equation for calculating centripetal force:

There’s another equation that relates the rotational speed (w) with the velocity like this:

Here’s a cool experiment you can do that will really show you how objects that move in a circle experience centripetal force. You can lift at least 10 balls by using only one! All you need are balls, fishing line or dental floss, and an old pen. Let’s calculate the velocity of the above experiment using our new circular motion equations. Let’s say you timed yourself, and you can get one ball to lift five identical balls when the one ball swings around once every second. Let’s calculate the acceleration, force, and speed. The net force acting on the ball is directed inwards. There might be more than one force acting on an object moving in a circle, but it’s the net force that adds them all up. The net force is proportional to the square of the speed (look at the equations again!). So if your speed increases three times, then the force increases by a factor of nine.


CIRCULAR MOTION WITH A CAR

Let’s do a sample problem involving a car:


CIRCULAR MOTION WITH A KID

Now let’s do a similar problem but this time with a kid instead of a car:


FAVORITE AMUSEMENT PARK RIDE

I love amusement park rides, even though I know what’s going on from the science side of things! Here’s one that’s always been a favorite of mine:


CIRCULAR MOTION AND FRICTION

Before we go any further, we need to take a look at how friction gets handled in these types of problems:


SWINGING BUCKET

You’ve come so far with your analysis that I really want to give you the “real way” to solve these types of problems. Normally, this method isn’t introduced to you until your second year in college, and that’s only if you’re an engineer taking Statics and Dynamics classes (the next level after this course). Here’s a step-by-step method that really puts all the pieces we’ve been working on all together into one:

Do you see how easy that is? It puts the FBD (free body diagram) together with the MAD (mass-acceleration diagram) and uses Newton’s Laws to solve for the things you need to know. Using pictures and equations, you can solve anything now!


CLOTHOID LOOPS

Let’s do something fun now… want to know about the physics of real roller coasters? It’s really important to know how much centrifugal force people experience, whether it’s in cars or roller coasters! In fact roller coaster loops used to be circular, but now they use clothoid loops instead to keep passengers happy during their ride so they don’t need nearly the acceleration that they’d need for a circular loop (which means less g-force so passengers don’t black out).


ROLLER COASTER MANEUVERS

There’s two main types of maneuvers a roller coaster: camel-backs and loops.


ROLLER COASTER ACTIVITY

Roller coasters are fun not just for their speed, but for their feelings of weightlessness one second and heaviness the next. One minute we’re rising up quickly, the next in free fall, then shoved against the door. It’s all about acceleration! The sections of a roller coaster you can approximate as circular motion include loops, small hills and dips (called camel backs), and banked turns. Now it’s time to do an experiment! All you need is a handful of marbles, some masking tape, and pipe insulation from the hardware store:


Roller coasters

Overview: Marbles can teach us a lot about energy, especially while they zoom down a custom-made coaster track!Today you’ll learn how energy can be transferred from kinetic energy to potential energy and back again.

What to Learn: You’lldiscover important concepts about how potential energy is converted into motion energy.

Materials

Marbles

Masking tape

3/4″ pipe foam insulation (NOT neoprene and NOT the kind with built-in adhesive tape)

Lab Time

1. Check your insulation tube. One side should be cut, so turn the tubing over and cut the other side, so you have two halves of the tubing.

2. To join the track, put tape on the inside and backside of the tubing.

3. To make a loop or corkscrew, use a third piece of tape to wrap all the way around the tube.

4. Tape one end of the tube as high as you can reach on the wall to start the marble rolling. Once you’re ready, release the marble and watch it fly down the track!


Observations

Where is the marble going the fastest?

When does the marble seem to slow down?

Why doesn’t the marble fly off the track when it goes upside-down?

To make the roller coasters, you’ll need foam pipe insulation, which is sold by the six-foot increments at

the hardware store. You’ll be slicing them in half lengthwise, so each piece makes twelve feet of track. It

comes in all sizes, so bring your marbles when you select the size. The ¾” size fits most marbles, but if

you’re using ball bearings or shooter marbles, try those out at the store. (At the very least you’ll get

smiles and interest from the hardware store sales people.) Cut most of the track lengthwise (the hard

way) with scissors. You’ll find it is already sliced on one side, so this makes your task easier. Leave a few

pieces uncut to become “tunnels” for later roller coasters.

Reading

You might have heard how energy cannot be created or destroyed, but it can be transferred or

transformed (if you haven’t, that’s okay – you’ll pick it up while doing this activity). We will observe two

types of energy here today: kinetic energy, and potential energy.

Kinetic energyis the energy of motion that an object has when it is pushed, flies, or falls.

Potential energyis the energy that an object has in relation to the system in which it exists. To imagine

this,pretend that you are shooting a bow and arrow. When you pull your arm back, the arrow doesn’t

have kinetic energy, because it isn’t moving. Yet the system has given it a lot of energy so that when

you release your fingers, the arrow will fly fast and far.

Roller coasters are a prime example of energy transfer: You start at the top of a big hill at low speeds

(high gravitational potential energy), then race down a slope at breakneck speed (potential

transforming into kinetic) until you bottom out and enter a loop (highest kinetic energy, lowest

potential energy). At the top of the loop, your speed slows (increasing your potential energy), but then


you speed up again and you zoom near the bottom exit of the loop (increasing your kinetic energy), and

you’re off again!

Exercises Answer the questions below:

1. What type of energy does a marble have while flying down the track of a roller coaster?

2. What type of energy does the marble have when you are holding it at the top of the track?

3. At the top of a camel back hill, which is higher for the marble, kinetic or potential energy?

4. At the top of an inverted loop, which energy is higher, kinetic or potential energy?


Answers to Exercises: Roller Coasters

1. What type of energy does a marble have while flying down the track of a roller coaster? (kinetic)

2. What type of energy does the marble have when you are holding it at the top of the track? (potential)

3. At the top of a camel back hill, which is higher for the marble, kinetic or potential energy? (potential)

4. At the top of an inverted loop, which energy is higher, kinetic or potential energy? (potential)


SPEED SKATING

You can find circular motion everywhere, including football, car racing, ice skating, and baseball. An ice

skater spins on ice, or a competition speed skater makes a turn… they are both examples of circular

motion. A turn happens when there’s a force component directed inward from the circular path. Let me

show you a couple of examples:


UNBANKED CAR TURN WITHOUT FRICTION

What about your car along a circular path? Let’s take a look at two different examples. The first is an

unbanked turn with friction:


BANKED CAR TURN WITH FRICTION

The second example is a banked turn, like NASCAR racing:


UNIVERSAL GRAVITATION

Gravity is the reason behind books being dropped and suitcases feeling heavy. It’s also the reason our atmosphere sticks around and oceans staying put on the surface of the earth. Gravity is what pulls it all together, and we’re going to look deeper into what this one-way attractive force is all about. Galileo was actually one of the first people to do science experiment on gravity. Galileo soon figured out that objects could be the same shape and different weights (think of a golf ball and a ping pong ball), and they will still fall the same. It was only how they interacted with the air that caused the fall rate to change. By studying ramps (and not just dropping things), he could measure how long things took to drop using not a stopwatch but a water clock (imagine having a sink that regularly dripped once per second). Whenever I teach a class about gravity, I’ll drop something (usually something large). After the heads whip around, I ask the hard question: “Why did it fall?” You already know the answer – gravity. But why? Why does gravity pull things down, not up? And when did people first start noticing that we stick to the surface of the planet and not float up into the sky? No one can tell you why gravity is – that’s just the way the universe is wired. Gravitation is a natural thing that happens when you have mass. Would it sound strange to you if I said that gravity propagates at the speed of light? If we suddenly made the sun disappear, the Earth’s orbit wouldn’t be instantaneously affected… it would take time for that information to travel to the earth. What does that mean? By the end of this section, you’ll be able to tell me about it. Let’s get started! So far, saying the force of gravity is pretty comfortable. When you throw a ball high in the air, the force of gravity slows it down and as it falls back to the earth the force of gravity speeds the ball up. The force of gravity causes an acceleration during this flight, and is called the acceleration of gravity. The acceleration of gravity g is the acceleration experienced by an object when the only force acting on it is the force of gravity. This value of g is the same no matter how massive the object is. It’s always 9.81 m/s2. Johannes Kepler, a German mathematician and astronomer in the 1600s, was one of the key players of his time in astronomy. Among his best discoveries was the development of three laws of planetary orbits. He worked for Tycho Brahe, who had logged huge volumes of astronomical data, which was later passed onto to Kepler. Kepler took this information to design and develop his ideas about the movements of the planets around the Sun. We’re going to go into deeper discussion about Kepler’s Laws in the next section, but here they are in a nutshell:

The Law of Orbits: All planets move in elliptical orbits, with the sun at one focus. The Law of Areas: A line joining the planet to the sun sweeps out equal areas in equal times. The Law of Periods: The square of the period of any planet is proportional tot he cube of the semi-

major axis of its orbit.

Did you notice that while Kepler’s Laws describe the motion of the planets around the sun, they don’t say why these paths are there? Kepler only hinted at an interaction between the sun and the planets to drive their motion, but not between the planets themselves, and it really was only a teensy hint.


Newton wasn’t satisfied with this explanation. He was determined to figure out the cause for the elliptical motion, especially since it wasn’t a circle or a straight line (remember Newton’s First Law: Objects in motion tend to stay in motion unless acted upon by an unbalanced force?) And circular motion needs centripetal force to keep the object following a curved path. So what force was keeping the planets in an ellipse around the sun?


INVERSE SQUARE LAW

One of Newton’s biggest contributions was figuring out how to show that gravity was the same force that

caused both objects like an apple to fall to the earth at a rate of 9.81 m/s2 AND the moon being

accelerated toward the earth but at a different rate of 0.00272 m/s2. If these are both due to the same

force of gravity, why are they different numbers then?

Why is the acceleration of the moon 1/3600th the acceleration of objects near the surface of the earth? It

has to do with the fact that gravity decreases the further you are from an object. The moon is in orbit

about 60 times further from the earth’s center than an object on the surface of the earth, which indicates

that gravity is proportional to the inverse of the square of the distance (also called the inverse square

law).

So the force of gravity acts between any two objects and is inversely proportional to the square of the

distance between the two centers. The further apart the objects are, the less they force of gravity is

between the two of them. If you separate the objects by twice the distance, the gravitational force goes

down by a factor of 4.


APPLYING THE INVERSE SQUARE LAW

All objects are attracted to each other with a gravitational force. You need objects the size of planets in

order to detect this force, but everything, everywhere has a gravitational field and force associated with

it. If you have mass, you have a gravitational attractive force. Newton’s Universal Law of Gravitation is

amazing not because he figured out the relationship between mass, distance, and gravitational force

(which is pretty incredible in its own right), but the fact that it’s universal, meaning that this applies

to every object, everywhere.

Newton suggested that every particle everywhere attracts every other particle with a force given by the following equation. If you have mass, then this force applies to you. Newton’s Universal Law of Gravitation is: That G is called the universal gravitation constant and is determined by doing experiments.


CAVENDISH EXPERIMENT

Lord Henry Cavendish in 1798 (about a century after Newton) performed experiments with a torsion balance to figure out the value of G. It’s a very small number, so Cavendish had to carefully calibrate his experiment! The reason the number is so small is because we don’t see the effects of gravity until objects are very massive, like a moon or a planet in size. Cavendish used an experiment where two small lead spheres were fastened to the ends of a rod which had a very fine string (actually a quartz fiber) attached to the middle so it could be lifted off the ground. This is called a torsion balance, meaning that you can carefully measure the twist in the string by measuring how much the rod spins around. (Torsion balances can be made from other materials that have a stiffer spring constant value, like metal rods.) Back to his experiment: Cavendish placed two large lead spheres next to the smaller spheres, which moved the larger spheres and exerted a torque on the rod, and Cavendish was able to calculate the value of G. The value of G is always the same, everywhere you go and any situation you apply it to. Once you know the masses and distances between objects, you can always calculate the force due to gravity with this one equation. Although Newton’s Law of Gravitation applies only to particles, you can apply it to real objects as long as the sizes of the objects is small when you compare it to the distances between them. You can concentrate an objects mass by shrinking it down to a particle using the idea of the center of mass like this: Let’s practice it now…


STAR WOBBLE

Today, scientists use Newton’s Law of Universal Gravitation to study the orbit of plants, moons, comets, asteroids, and even galaxies, like this: By calculating how each object in the solar system “yanks” on each other, we can make precise predictions about where to expect Jupiter to be next month, or when a comet will graze the sun, or if an asteroid is too close for comfort. Before this equation existed, astronomers weren’t sure why at certain times Jupiter would wobble a little in orbit… but now we know it’s because Saturn was close enough to yank hard enough on Jupiter for us to see its effects.


Star Wobble

Introduction: How do astronomers find planets around distant stars? If you look at a star through

binocularsor a telescope, you’ll quickly notice how bright the star is, and how difficult it is to see anything

other than the star, especially a small planet that doesn’t generate any light of its own! Astronomers look for

a shift, or wobble, of the star as it gets gravitationally “yanked” around by the orbiting planets. By measuring

this wobble, astronomers can estimate the size and distance of larger orbiting objects.

Doppler spectroscopy is one way astronomers find planets around distant stars. If you recall the lesson where

we created our own solar system in a computer simulation, you remember how the star could be influenced

by a smaller planet enough to have a tiny orbit of its own. This tiny orbit is what astronomers are trying to de-

tect with this method.

In this lab exercise, you're going to observe how different sizes of planets and stars wobble when they are con-

nected by gravity, or in our case, a toothpick!

Materials:

5 bouncy balls of different sizes and weights (soft enough to stab with a toothpick)

Scale

Toothpicks Procedure:

1. Does your ball have a number written on it? If so, that’s the weight, and you can skip measuring the

mass with a scale.

2. If not, measure the mass of each one and make a note in the data table on the next page.

3. Take the heaviest ball and spin it on the table (by itself). Can you get it to spin in place? That’s like a

rotating Sun without any planets around it.

4. Next, you're going to be creating combinations of stars and planets using your toothpicks. Following

the combinations listed in the table on the next page, insert a toothpick into the heavier ball, then in-

sert the other end into the lighter ball (this will mimic gravity connecting the two).

5. Now spin the heavier of the two balls and observe/record how much it wobbles compared to the

other combinations in the space provided.


Combination

Wobble Observations

Ball 1 and Ball 2

Ball 1 and Ball 3

Ball 1 and Ball 4

Ball 2 and Ball 3

Ball 2 and Ball 4

Ball 3 and Ball 4

Which combination had the most wobble? Which combination had the least wobble? You should have seen that the combination with the least wobble was the pairing of the heaviest ball and the lightest ball, and the combination with the most wobble was the pairing of the two balls with the most similar masses. But why is this?

Well when you spun the unconnected ball in the beginning, it was allowed to spin in place because its center

of mass was directly in the center of the ball. However, when you connect two balls together by a toothpick,

the center of mass is shifted away from the axis of the heavier ball. When the two are spun, they rotate on

an axis around their center of mass. So the further away the center of mass is from the heavier ball, the more

wobble you will see! Check out the diagrams and equations on the next page.

Ball

Mass(grams)

1

2

3

4


In the diagram above, you can see how the center of mass of the two balls is not centered on the axis of

the larger ball, so when spun, it will wobble. The equations that relate this system are as follows:

m1 d1 = m2 d2

d1 + d2 = R

So imagine you have two balls of equal mass (m). This would mean d1= d2, and the center of mass would be in

the center of the toothpick, causing the most wobble possible!


Problems:

1. If you have two balls; one with mass 10 kg, one with mass 2 kg that are separated by a 2 meter

rod, how far from the center of the 10 kg ball with the center of mass be?

2. How far will the center of mass be from the 2 kg ball?

3. Given that Mercury (the closest planet to the sun) has a mass of 328.5x1021 kg, the sun has a mass of

1.989x1030 kg, and that the distance between the two bodies is 57,910,000 km, what is the distance from the center of the sun to the center of mass of the two bodies?


Answers:

1. 0.33 m

2. 0.67 m

3. Only 9.7 km!


IF THE EARTH GAINED WEIGHT, WOULD YOU?

Weight is nothing more than a measure of how much gravity is pulling on you. This is why you can be “weightless” in space. You are still made of stuff, but there’s no gravity to pull on you so you have no weight. The larger a body is, the more gravitational pull (or in other words the larger a gravitational field) it will have. The Moon has a fairly small gravitational field (if you weighed 100 pounds on Earth, you’d only be 17 pounds on the Moon). The Earth’s field is fairly large and the Sun has a HUGE gravitational field (if you weighed 100 pounds on Earth, you’d weigh 2,500 pounds on the Sun!). As a matter of fact, the dog and I both have gravitational fields! Since we are both bodies of mass, we have a gravitational field which will pull things toward us. All bodies have a gravitational field. However, my mass is so small that the gravitational field I have is miniscule. Something has to be very massive before it has a gravitational field that noticeably attracts another body. So what’s the measurement for how much stuff you’re made of? Mass. Mass is basically a weightless measure of how much matter makes you you. A hamster is made of a fairly small amount of stuff, so she has a small mass. I am made of more stuff, so my mass is greater than the hamster’s. Your house is made of even more stuff, so its mass is greater still. So, here’s a question. If you are “weightless” in space, do you still have mass? Yes, the amount of stuff you’re made of is the same on Earth as it is in your space ship. Mass does not change, but since weight is a measure for how much gravity is pulling on you, weight will change.


TURNING THE SUN INTO A BLACK HOLE

At some point in the future you may ask yourself this question, “How can gravity pull harder (put more

force on some things, like bowling balls) and yet accelerate all things equally?”

Now that we have studied Newton’s laws, you can see that the above statement doesn’t make any sense

at all! More force equals more acceleration is basically Newton’s Second law. The explanation for this is

inertia.


GRAVITY AND INERTIA

Inertia is basically how much force is needed to get something to move or stop moving. Now, let’s get

back to gravity and acceleration. Let’s take a look at a bowling ball and a golf ball.

Gravity puts more force on the bowling ball than on the golf ball. So the bowling ball should accelerate faster since there’s more force on it. However, the bowling ball is heavier so it is harder to get it moving. Vice versa, the golf ball has less force pulling on it but it’s easier to get moving. Do you see it? The force and inertia thing equal out so that all things accelerate due to gravity at the same rate of speed! Gravity had to be one of the first scientific discoveries. Whoever the first guy was to drop a rock on his foot, probably realized that things fall down! However, even though we have known about gravity for many years, it still remains one of the most elusive mysteries of science. At this point, nobody knows what makes things move toward a body of mass. Why did the rock drop toward the Earth and on that guy’s foot? We still don’t know. We know that it does, but we don’t know what causes a gravitational attraction between objects. Gravity is also a very weak force. Compared to magnetic forces and electrostatic forces, the gravitational force is extremely weak. How come? No one knows. A large amount of amazing brain power is being used to discover these mysteries of gravity. Maybe it will be you who figures this out!


Watch Your Weight Overview: If you could stand on the Sun without being roasted, how much would you weigh? The gravitational

pullis different for different objects. Let’s find out which celestial object you’d crack the pavement on, and which your lightweight toes would have to be careful about jumping on in case you leapt off the planet.

What to Learn: Weight is nothing more than a measure of how much gravity is pulling on you. Mass is a

measureof how much stuff you’re made out of. Weight can change depending on the gravitational field you are

standing in. Mass can only change if you lose an arm.

Materials

Scale to weigh yourself

Calculator

Pencil

Experiment

1. We need to talk about the difference between weight and mass. In everyday language, weight and mass are used interchangeably, but scientists know better.

2. Mass is how much stuff something is made out of. If you’re holding a bowling ball, you’ll notice that it’s hard to get started, and once it gets moving, it needs another push to get it to stop. If you leave the bowling

ball on the floor, it stays put. Once you push it, it wants to stay moving. This “sluggishness” is called inertia.

Mass is how much inertia an object has.

3. Every object with mass also has a gravitational field, and is attracted to everything else that has mass. The amount of gravity something has depends on how far apart the objects are. When you step on a bathroom

scale, you are reading your weight, or how much attraction is between you and the Earth.

4. If you stepped on a scale in a spaceship that is parked from any planets, moons, black holes, or other objects, it would read zero. But is your mass zero? No way. You’re still made of the same stuff you were

on Earth, so your mass is the same. But you’d have no weight.

5. What is your weight on Earth? Let’s find out now.

6. Step on the scale and read the number. Write it down.

7. Now, what is your weight on the Moon? The correction factor is 0.17. So multiply your weight by 0.17 to find what the scale would read on the Moon.

8. For example, if I weigh 100 pounds on Earth, then I’d weight only 17 pounds on the Moon. If the scale reads

10 kg on Earth, then it would read 1.7 kg on the Moon. Complete the data table..


Watch Your Weight Data Table

Weight on Planet/Object = Weight on Earth x Gravity Correction

Planet/Object Weight on Earth Gravity Correction

Weight on

Planet/Object

The Sun 28

Mercury 0.38

Venus 0.91

Earth 1

Moon 0.17

Mars 0.38

Jupiter 2.14

Saturn 0.91

Uranus 0.86

Neptune 1.1

Pluto 0.08

Outer Space 0


Betelgeuse 14,000

White Dwarf 1,300,000

Neutron Star (Pulsar) 140,000,000,000

Black Hole ∞

Reading

Weight is nothing more than a measure of how much gravity is pulling on you. This is why you can be “weightless”

in space. You are still made of stuff, but there’s no gravity to pull on you so you have no weight. The larger a body

is, the more gravitational pull (or in other words the larger a gravitational field) it will have.

The Moon has a fairly small gravitational field (if you weighed 100 pounds on Earth, you’d only be 17 pounds on

the Moon).The Earth’s field is fairly large and the Sun has a HUGE gravitational field (if you weighed 100 pounds

on Earth, you’d weigh 2,500 pounds on the Sun!).

As a matter of fact, the dog and I both have gravitational fields! Since we are both bodies of mass, we have a

gravitational field which will pull things toward us. All bodies have a gravitational field. However, my mass is so

small that the gravitational field I have is miniscule. Something has to be very massive before it has a

gravitational field that noticeably attracts another body.

So what’s the measurement for how much stuff you’re made of? Mass. Mass is basically a weightless measure of

how much matter makes you you. A hamster is made of a fairly small amount of stuff, so she has a small mass. I am

made of more stuff, so my mass is greater than the hamster’s. Your house is made of even more stuff, so its mass is

greater still. So, here’s a question. If you are “weightless” in space, do you still have mass? Yes, the amount of stuff

you’re made of is the same on Earth as it is in your space ship. Mass does not change, but since weight is a measure

for how much gravity is pulling on you, weight will change.

Did you notice that I put weightless in quotation marks? Wonder why?

Weightlessness is a myth! Believe it or not, one is never weightless. A person can be pretty close to weightless in very deep space, but the astronauts in a space ship actually do have a bit of weight.

Think about it for a second. If a space ship is orbiting the Earth, what is it doing? It’s constantly falling! If it wasn’t moving forward at tens of thousands of miles an hour it would hit the Earth. It’s moving fast enough to fall

around the curvature of the Earth as it falls but, indeed, it’s falling as the Earth’s gravity is pulling it to us.

Otherwise the ship would float out to space. So what is the astronaut doing? She’s falling, too! The astronaut and

the space ship are both falling to the Earth at the same rate of speed and so the astronaut feels weightless in

space. If you were in an elevator and the cable snapped, you and the elevator would fall to the Earth at the same

rate of speed. You’d feel weightless! (Don’t try this at home!)


Either now, or at some point in the future you may ask yourself this question, “How can gravity pull harder (put

more force on some things, like bowling balls) and yet accelerate all things equally?” When we get into

Newton’s laws in a few lessons, you’ll realize that doesn’t make any sense at all. More force equals more

acceleration is basically Newton’s Second law.

Well, I don’t want to take too much time here since this is a little deeper then we need to go but I do feel some

explanation is in order to avoid future confusion. The explanation for this is inertia. When we get to Newton’s First law

we will discuss inertia. Inertia is basically how much force is needed to get something to move or stop moving.

Now, let’s get back to gravity and acceleration. Let’s take a look at a bowling ball and a golf ball. Gravity puts more

force on the bowling ball than on the golf ball. So the bowling ball should accelerate faster since there’s more

force on it. However, the bowling ball is heavier so it is harder to get it moving. Vice versa, the golf ball has less

force pulling on it but it’s easier to get moving. Do you see it? The force and inertia thing equal out so that all

things accelerate due to gravity at the same rate of speed!

Gravity had to be one of the first scientific discoveries. Whoever the first guy was to drop a rock on his foot,

probably realized that things fall down! However, even though we have known about gravity for many years, it

still remains one of the most elusive mysteries of science. At this point, nobody knows what makes things move

toward a body of mass.

Why did the rock drop toward the Earth and on that guy’s foot? We still don’t know. We know that it does, but we

don’t know what causes a gravitational attraction between objects. Gravity is also a very weak force. Compared to

magnetic forces and electrostatic forces, the gravitational force is extremely weak. How come? No one knows. A

large amount of amazing brain power is being used to discover these mysteries of gravity. Maybe it will be you

who figures this out!

Exercises

1. Of the following objects, which ones are attracted to one another by gravity?

a) Apple and Banana b) Beagle and Chihuahua c) Earth and You d) All of the above

2. True or False: Gravity accelerates all things differently

3. True or False: Gravity pulls on all things differently


4. If I drop a golf ball and a golf cart at the same time from the same height, which hits the ground first?

5. There is a monkey hanging on the branch of a tree. A wildlife biologist wants to shoot a tranquilizer dart at

the monkey to mark and study him. The biologist very carefully aims directly at the shoulder of the

monkey and fires. However, the gun makes a loud enough noise that the monkey gets scared, lets go of the

branch and falls directly downward. Does the dart hit where the biologist was aiming, or does it go higher

or lower then he aimed? (This, by the way, is an old thought problem.)


Answers to Exercises: Watch Your Weight

1. Of the following objects, which ones are attracted to one another by gravity?

1. Apple and Banana b) Beagle and Chihuahua c) Earth and You d) All of the above

2. True or False: Gravity accelerates all things differently

3. True or False: Gravity pulls on all things differently

4. If I drop a golf ball and a golf cart at the same time from the same height, which hits the ground first? (They both hit the ground at the same time.)

5.There is a monkey hanging on the branch of a tree. A wildlife biologist wants to shoot a tranquilizer dart at the

monkey to mark and study him. The biologist very carefully aims directly at the shoulder of the monkey and fires.

However, the gun makes a loud enough noise that the monkey gets scared, lets go of the branch and falls directly

downward. Does the dart hit where the biologist was aiming or does it go higher or lower then he aimed? (The

monkey and the dart fall downward at the same rate of speed. So the dart would hit exactly where the biologist

aimed! In fact, if the monkey didn’t let go, the dart would have hit lower than the biologist aimed.)


ROCKETS AND GRAVITY

What happens to gravity if you’re in a rocket moving up through the atmosphere to a satellite in orbit? Does it remain the same or does it change during your flight? Here’s information about the earth:

Radius: 3,959 miles (6,371 km) Distance from Sun: 92,960,000 miles (149,600,000 km) Mass: 5.972E24 kg

The International Space Station, an object about 72 meters long by 108 meters wide and 20 meters high stays in an orbital altitude of between 330 km (205 miles) and 410 km (255 miles), and moves at a rate of 27,724 kph(17,227 mph).


GRAVITY FOR DIFFERENT OBJECTS

First we’re going to assume the earth is like a ball in that it’s a perfect sphere, and also that the density of the earth is even and it depends only how far from the center of the earth you are. Let’s also assume the earth isn’t rotating. Once we have these things in mind, then the magnitude of the force of gravity acting on an object goes like this…


REAL ROCKET LAUNCH

STS-119 (ISS assembly flight 15A) was a space shuttle mission to the International Space Station (ISS)

which was flown by Space Shuttle Discovery during March 2009. It delivered and assembled the fourth

starboard Integrated Truss Segment (S6), and the fourth set of solar arrays and batteries to the station.

The launch took place on March 15, 2009, at 7:43 p.m. EDT. Discovery successfully landed on March 28,

2009, at 3:13 p.m. EDT.

Here’s the ISS (International Space Station) from orbit. Many wonders are visible when flying over the Earth at night, especially if you are an astronaut on the International Space Station (ISS)! Passing below are white clouds, orange city lights, lightning flashes in thunderstorms, and dark blue seas. On the horizon is the golden haze of Earth’s thin atmosphere, frequently decorated by dancing auroras as the video progresses. The green parts of auroras typically remain below the space station, but the station flies right through the red and purple auroral peaks. You’ll also see solar panels of the ISS around the frame edges. The wave of approaching brightness at the end of each sequence is just the dawn of the sunlit half of Earth, a dawn that occurs every 90 minutes, as the ISS travels at 5 miles per second to keep from crashing into the earth.


THE EARTH’S CRUST IS NOT UNIFORM

There are three main differences between assuming the earth is round, uniformly dense, and not rotating as we did before. First, the crust is not uniform. There are lumps and clumps everywhere that vary the density add up to make small variations in the force of gravity that we can actually measure with objects in free-fall motion. It’s actually how scientists find pockets of oil in the earth. They measure the surface gravity and plot it out, and if there’s a large enough deviation, it means there’s something interesting underground.

This image of the Mors salt dome in Denmark was studied for radioactive waste disposal. It’s a surface

gravity survey that measures the acceleration due to gravity that shows something interesting is

underground! The dots are the places where gravity was actually measured. The unit of measurement for

these deviations is called the “milligal” for Galileo, where 1 gal = 1,000 mgal = 1 cm/s3.


THE EARTH IS NOT A SPHERE

The second problem with our assumption sis that the earth is not a sphere. It’s flattened a bit at the poles and bulges out at the equator. The ring around equator is larger than a ring around the poles by 21 km, which makes the poles closer to the center of the earth than the equator! Free-fall at the poles is slightly more than free-fall at the equator. But before you book a trip to skydive in Ecuador, Colombia, Brazil, Sao Tome, Gabon, the Republic of the Congo, the Democratic Republic of the Congo, Uganda, Kenya, Somalia, Maldives, Indonesia or Kiribati, let’s talk about the assumption we made… The earth really does rotate. That’s not a surprise. How does this affect the value of g then? The bottom line is that gravity changes with altitude from 9.78 to 9.84 m/s2., mostly due to the earth spinning, but some to the earth not being a perfect sphere. Here are the Scientific Concepts to remember about Gravity:

Gravity is a force that attracts things to one another. All bodies (objects) have a gravitational field. The larger a body is, the greater the strength of the gravitational field. Bodies must be very, very large before they exert any noticeable gravitational field. Gravity accelerates all things equally. Which means all things speed up the same amount as they

fall. Gravity does not care what size things are or whether things are moving. All things are accelerated

towards the Earth at the same rate of speed. Gravity does pull on things differently. Gravity is pulling greater on objects that weigh more. Weight is a measure of how much gravity is pulling on an object. Mass is a measure of how much matter (how many atoms) make up an object.


PLANETARY AND SATELLITE MOTION

Johannes Kepler, a German mathematician and astronomer in the 1600s, was one of the key players of his time in astronomy. Among his best discoveries was the development of three laws of planetary orbits. He worked for Tycho Brahe, who had logged huge volumes of astronomical data, which was later passed onto to Kepler. Kepler took this information to design and develop his ideas about the movements of the planets around the Sun. Kepler’s 1st Law states that planetary orbits about the Sun are not circles, but rather ellipses. The Sun lies at one of the foci of the ellipse.Well, almost. Newton’s Laws of Motion state that the Sun can’t be stationary, because the Sun is pulling on the planet just as hard as the planet is pulling on the Sun. They are yanking on each other. The planet will move more due to this pulling because it is less massive. The real trick to understanding this law is that both objects orbit around a common point that is the center of mass for both objects. If you’ve ever swung a heavy bag of oranges around in a circle, you know that you have to lean back a bit to balance yourself as you swing around and around. It’s the same principle, just on a smaller scale. In our solar system the Sun has 99.85% of the mass, so the center of mass between the Sun and any other object actually lies inside the Sun (although not at the center). Kepler’s 2nd Law states that a line connecting the Sun and an orbiting planet will sweep out equal areas in for a given amount of time. The planet’s speed decreases the further from the Sun it is located (actually, the speed varies inversely with the square‐root of the distance, but you needn’t worry about that). You can see this for yourself by tying a ball to the end of a string and whirl it around in a circle. After a few revolutions, let the string wind itself up around your finger. As the string length shortens, the ball speeds up. As the planet moves inward, the planet’s orbital speed increases. Embedded in the second law are two very important laws: conservation of angular moment and conservation of energy. Although those laws might sound scary, they are not difficult to understand. Angular momentum is distance multiplied by mass multiplied by speed. The angular momentum for one case must be the same for the second case (otherwise it wouldn’t be conserved). As the planet moves in closer to the Sun, the distance decreases. The speed it orbits the Sun must increase because the mass doesn’t change. Just like you saw when you wound the ball around your finger. Energy is the sum of both the kinetic (moving) energy and the potential energy (this is the “could” energy, as in a ball dropped from a tower has more potential energy than a ball on the ground, because it “could” move if released). For conservation of energy, as the planet’s distance from the Sun increases, so does the gravitational potential energy. Again, since the energy for the first case must equal the energy from the second case (that’s what conservation means), the kinetic energy must decrease in order to keep the total energy sum a constant value. Kepler’s 3rd Law is an equation that relates the revolution period with the average orbit speed. The important thing to note here is that mass was not originally in this equation. Newton came along shortly after and did add in the total mass of the system, which fixed the small error with the equation. This makes sense, as you might imagine a Sun twice the size would cause the Earth to orbit faster. However, if we double the mass of the Earth, it does not affect the speed with which it orbits the Sun. Why not? Because the Earth is soooo much smaller than the Sun that increasing a planet’s size generally doesn’t make a difference in the orbital speed. If you’re working with two objects about the same size, of course, then changing one of the masses absolutely has an effect on the other.


FUN ACTIVITY WITH KEPLER’S LAWS

Kepler’s Laws of planetary orbits explain why the planets move at the speeds they do. You’ll be making a

scale model of the solar system and tracking orbital speeds with this next easy experiment. All you need

is a long measuring tape or length of string and some tape to mark the planet positions. A bunch of

friends is helpful, also!


Kepler's Swinging System

Introduction:

Kepler’s Laws of planetary orbits explain why the planets move at the speeds they do. You’ll be making a

scale model of the solar system and tracking orbital speeds.

Kepler’s 1st Law states that planetary orbits about the Sun are not circles, but rather ellipses. The Sun lies

at one of the foci of the ellipse. Kepler’s 2nd Law states that a line connecting the Sun and an orbiting

planet will

sweep out equal areas in for a given amount of time. Translation: the further away a planet is from the

Sun, the slower it goes.

In this experiment, you're going to be making a scale model of our solar system. You'll need at least one

partner, but more partners are better. Each person will represent the Sun or a planet in the solar system!

Materials:

30 Foot Measuring Tape (or 30 feet of rope)

2 – 10 People

Large area or field

Stopwatch

Procedure:

1. Begin with one person holding onto the 30 foot tape measure (they will be the Sun).

2. Have another person grab the end of the tape measure and start walking away (they will be Pluto).

3. Have Pluto keep walking away until the Sun sees the 7 foot mark. At this point, Neptune should

grab onto the tape measure at the sun, and keep walking with Pluto.

4. At 15' 4”, Uranus should grab hold as did Neptune and keep walking.

5. At 22' 8”, Saturn should grab hold.

6. At 26', Jupiter grabs hold.

7. At 28' 10”, Mars grabs hold.

8. At 29' 3”, Earth grabs hold.

9. At 29' 6”, Venus grabs hold.

10. Finally at 29' 8”, Mercury grabs hold.

11. Make sure the measuring tape is taught. You have now made a scale model of the solar system!

12. Next, grab your stopwatch and have everyone except Pluto and the Sun let go of the measuring

tape (but don't let them move!).

13. Have the Sun stand in the same place, and have Pluto walk around the Sun at a constant steady

pace while keeping the measuring tape taught.

14. Time how long it takes Pluto to make one complete revolution and record it in the table below.

15. Next it's Uranus' turn. Have Uranus do the same thing as Pluto while maintaining the exact same

walking speed.

16. Once Uranus has made a complete rotation, record the time, and let Saturn take their turn.

17. Keep this process going until times for each planet have been recorded.


You should clearly see that it takes Pluto much longer to complete an orbit around the Sun than Mercury.

This is Kepler’s 2nd Law in action. The further away a planet is, the longer it takes to orbit!

This 30 foot model of the solar system is an incredibly small scale compared to the real thing. In fact, with

Pluto 30 feet from the Sun, you've made a 1 : 642,388,000,000 model! In reality, Pluto is about

3,650,000,000 miles from the Sun! Not only are the distances between the planets extraordinarily larger

than what you've created, the sizes of the individual planets is also way different. At this scale, the Sun

would be the size of a flea, and Pluto would be the size of a red blood cell!


Problems:

1. Salt flats are the result of bodies of water that have dried up over long periods of time. The

resulting areas are incredibly flat and can be very large. The Salar de Uyuni in Bolivia is the largest

salt flat on Earth. It has an area of 4,086 square miles, and only varies in elevation by around three

feet! If you were to go to Bolivia with a 50 mile long tape measure and had Pluto go 50 miles away

from the Sun, how far away would the Earth be from the Sun?

2. Since Pluto is actually 3,650,000,000 miles from the Sun, what scale would this model be? (Hint:

Take the ratio of the distances form the Sun)

3. On this scale, how far would someone representing Pluto have to walk to make a complete circle

around the Sun? (Hint: Try using the circumference of a circle C = 2 π r)


Answers:

1. About 2.2 miles

2. 1 : 73,000,000

3. 314 miles! Let's hope the don't have to walk!


ELLIPTICAL ORBITS

If one of Kepler’s Laws describe the orbits of satellites as being an elliptical orbit, you might be

wondering what an ellipse is! Here’s a really neat way to make an ellipse using a pencil and a rubber

band:


APPLYING KEPLER’S LAWS

Let’s try a few practice problems so you get more familiar with how to use and apply Kepler’s Laws to

real world physics problems…


JUPITER’S MOONS

A satellite is an object that orbits the sun, earth or other massive body like a planet, moon, asteroid, or

even galaxy. There are two kinds of satellites: natural, like the moon, and man-made, like the Hubble

Space Telescope.


NEW PLANETS

But physics doesn’t care if a satellite is man-made or not. The laws of physics and math equations still

apply no matter where the satellite came from.


WHY SATELLITES STAY IN ORBIT

An important concept to understand is that a satellite is a projectile, meaning that only the force of

gravity is acted on it (once it’s launched). In order to maintain it’s orbit, a satellite needs to fall

continuously at the same rate that the earth is curving away from it.


SATELLITE CRASH!

The Hubble Space Telescope (HST) zooms around the Earth once every 90 minutes (about 5 miles per second), and in August 2008, Hubble completed 100,000 orbits! Although the HST was not the first space telescope, is the one of the largest and most publicized scientific instrument around. Hubble is a collaboration project between NASA and the ESA (European Space Agency), and is one of NASA’s “Great Observatories” (others include Compton Gamma Ray Observatory, Chandra X-Ray Observatory, and Spitzer Space Telescope). Anyone can apply for time on the telescope (you do not need to be affiliated with any academic institution or company), but it’s a tight squeeze to get on the schedule. Here’s a neat experiment you can do with a sheet of paper and a marble to show how satellites need to move in order to stay in orbit around the earth.


Satellite Crash Introduction: The Hubble Space Telescope (HST) zooms around the Earth once every 90 minutes (about

5miles per second), and in August 2008, Hubble completed 100,000 orbits! Although the HST was not the

first space telescope, is the one of the largest and most publicized scientific instrument around. Hubble is a

collaboration project between NASA and the ESA (European Space Agency), and is one of NASA’s “Great

Observatories” (others include Compton Gamma Ray Observatory, Chandra X-Ray Observatory, and Spitzer

Space Telescope). Anyone can apply for time on the telescope (you do not need to be affiliated with any

academic institution or company), but it’s a tight squeeze to get on the schedule.

Hubble’s orbit zooms high in the upper atmosphere to steer clear of the obscuring haze of molecules in

the sea of air. Hubble’s orbit slowly decays over time and begins to spiral back into Earth until the

astronauts bump it back up into a higher orbit.

Materials:

Marble

Paper

Tape

Procedure:

1. Take your paper and roll it into a cone shape

2. Take a couple pieces of tape and secure the cone in place

3. Now take your marble and place it in the cone. Try to spin the cone at just the right speed so that

the marble doesn't fly out, or fall back into the hole!

4. Once you've had some practice, see if you can maintain the marble's path at a specific distance from the center hole. This is just like changing a satellite's orbital radius from Earth!

Unit 7: Lesson 2 Name ________________________


The marble rolling around the cone is a good analogy for a satellite orbiting Earth. When a satellite is put into

orbit, there are no big rockets on board that the satellite can use to maintain or change its orbit, so how

does it stay at the same distance from Earth, and not fall in due to gravity?

Well the satellite is effectively constantly free falling around Earth, and since in space there is no air to

create drag and slow it down, it maintains the same speed, and thus the same orbital radius.

Using the Hubble Space Telescope as an example, let's analyze the motion of a satellite orbiting Earth. In

this case we can assume that the Earth has a much larger mass than the satellite, which makes things easier.

The equation for the orbital velocity (v) of the satellite is given as follows:

Where v is the orbital velocity, G is the Gravitational Constant (6.67 x 10-11

m/kg2), M is the mass of the

Earth, and r is the radius form the center of the Earth to the orbit of the satellite.

So let's calculate the height above Earth's surface that Hubble orbits at. We know it's traveling around 5

miles per second (it's actually traveling 7,500 meters per second). We also know the mass of the Earth is 5.97

x 1024

kg, so what is r?

7,500 m/s = (6.67 x 10-11

m/kg2 * 5.97 x 10

24 kg / r )

1/2

56,250,000 m2/s

2 = 3.98 x 10

14 m/kg * r

r = 7.08x106 meters from the center of Earth

Earth has a radius of about 6,371,000 m, so Hubble is orbiting at a height of about 700 km above the

surface of the Earth.


Exercises:

1. Imagine a satellite is orbiting very far from Earth, with r = 550,000 km. How fast will the satellite

be orbiting if it is to maintain a circular orbit?

2. If we assume the Earth's mass is sufficiently smaller than the Sun's mass, how far from the center

of the Sun are we if we orbit at 30 km/s? (Hint: the mass of the Sun is 1.989x1030

kg)

3. How far from Earth would a satellite be if it were orbiting at 12,000 m/s?


Answers:

2. 850 m/s

3. 1.47x1011

m or 147 million km!

4. It's impossible! If no other forces are acting on the satellite, they would have to be below the surface

of the Earth!


SATURN’S MOONS

Kepler’s Law of Periods relates the period T of any planet around the sun to the cube of the semi-major axis a of the orbit:

If T is in earth years, a is in AU (1 AU = distance from the sun to the earth), and M = solar mass, then the equation above reduces to: T2 = a3.


ORBITS OF SATELLITES

A satellite is an object that does around a planet, a star, or other similar object. Here’s how you can figure out the net force of a satellite as well as the velocity of the satellite, since the only force applied to the satellite is from gravity:


ORBITAL MECHANICS

For a satellite orbiting around the earth at a given distance. What is the speed and acceleration of the

satellite? Do you think you need to know the masses of the satellite and the earth, or just one?


EQUATIONS FOR CIRCULAR MOTION

Here’s an overview of all the equations we’ve been using so far for our study in uniform circular motion:


HOW FAST IS THE MOON?

How fast does the moon travel around the earth? And how far away is the moon really? Here’s a way to

figure out!


CALLISTO

Callisto is one of Jupiter’s moons. Would it be really cool to be able to approximate the size of Jupiter

based on watching the motion of Callisto? For example, if you knew how long it took Callisto to orbit

around Jupiter, and the furthest distance it traveled away from it (both of which you could measure from

a backyard telescope)? Here’s how:


SPACE STATION SPEED

One of the questions I get asked a lot is how fast does the International Space Station travel around the

Earth? Here’s how to figure this out (hint: it’s a lot like the one we did previously with Jupiter and

Callisto!)

Finding the Mass of the Earth Now let’s estimate the mass of the Earth. You already know what it is

(because we’ve been using it a lot in our previous calculations), so let’s pretend we don’t know and figure

out first-hand.


HALLEY’S COMET

Comet Hally takes 76 years to orbit around the sun, and in 1986 it came as close as it possibly could to

the sun (89,000,000 km). Can we figure out how far the comet gets from the sun based on this

information? Sure! Here’s how:


WEIGHTLESSNESS

So, here’s a question. If you are “weightless” in space, do you still have mass? Yes, the amount of stuff you’re made of is the same on Earth as it is in your space ship. Mass does not change but since weight is a measure for how much gravity is pulling on you, weight will change. Did you notice that I put weightless in quotation marks? Wonder why? Weightlessness is a myth! Believe it or not, one is never weightless. A person can be pretty close to weightless in very deep space but the astronauts in a space ship actually do have a bit of weight. Think about it for a second. If a space ship is orbiting the Earth what is it doing? It’s constantly falling! If it wasn’t moving forward at 10’s of thousands of miles an hour it would hit the Earth. It’s moving fast enough to fall around the curvature of the Earth as it falls but, indeed, it’s falling as the Earth’s gravity is pulling it to us. Otherwise the ship would float out to space. So what is the astronaut doing? She’s falling too! The astronaut and the space ship are both falling to the Earth at the same rate of speed and so the astronaut feels weightless in space. If you were in an elevator and the cable snapped, you and the elevator would fall to the Earth at the same rate of speed. You’d feel weightless! (Don’t try this at home!)


PHYSICS FUN IN AN ELEVATOR

Now let’s have fun with an elevator and a bathroom scale, since we can’t easily jump ourselves into orbit.


BUST A MYTH

Now let’s bust the myth of weightlessness in space for good…


BINARY SYSTEM

A binary system exists when objects approach each other in size (and gravitational fields), the common

point they rotate around (called the center of mass) lies outside both objects and they orbit around each

other. Astronomers have found binary planets, binary stars, and even binary black holes. The path of a

planet around the Sun is due to the gravitational attraction between the Sun and the planet. This is true

for the path of the Moon around the Earth, and Titan around Saturn, and the rest of the planets that have

an orbiting moon. Here’s a neat experiment you can do with some things you already have around the

house to show you how binary planetary systems work:


Binary Planetary Systems Overview: A binary system exists when objects approach each other in size (and gravitational fields), the

commonpoint they rotate around (called the center of mass) lies outside both objects and they orbit around each

other. Astronomers have found binary planets, binary stars, and even binary black holes. Students will know that

the path of a planet around an object is due to the gravitational attraction between the object and the planet.

What to Learn: The path of a planet around the Sun is due to the gravitational attraction between the Sun and theplanet. This is true for the path of the Moon around the Earth, and Titan around Saturn, and the rest of the planets that have an orbiting moon.

Materials

Soup cans or plastic containers with holes punched (like plastic yogurt containers, butter tubs, etc.)

String

Water

Sand

Rocks

Pebbles

Baking soda

Vinegar

Experiment

1. Thread one end of the string through one of the holes and tie a strong knot. Really strong.

2. Tie the other end through the other hole and tie off.

3. Go outside.

4. Fill your can partway with water.

5. Move away from everyone before you start to swing your container in a gentle circle. As you spin faster and faster, notice where the water is inside the container.

6. Now empty out the water and replace it with rocks. Spin again and fill out the data table.

7. To make carbon dioxide gas, you’ll need to work with another lab team. Cover the bottom of your container

with baking soda. Add enough vinegar so that the bubbles reach the top without overflowing. Wait

patiently for the bubbles to subside. You now have a container filled with carbon dioxide gas (and a little

sodium acetate, the leftovers from the reaction). Carefully pour this into the empty container from the

other lab team. They can spin again and record their results. When they are done, borrow their container

and give them yours so they can fill it for you.


Binary Planetary Systems Data Table

When filling out the third column, notice how hard or easy it was to spin the container, what it felt like, which way it faced, etc. Record everything you can about each one.

Item in the Can State of Matter What did you notice?

(solid, liquid, or gas?)

Water

Rocks

Sand

Air

Pebbles

Carbon Dioxide Gas

Reading


The path of a planet around the Sun is due to the gravitational attraction between the Sun and the

planet. This is true for the path of the Moon around the Earth, and Titan around Saturn, and the

rest of the planets that have an orbiting moon.

Charon and Pluto orbit around each other due to their gravitational attraction to each other.

However, Charon is not the moon of Pluto, as originally thought. Pluto and Charon actually orbit

around each other. Pluto and Charonalso are tidally locked, just like the Earth-Moon system,

meaning that one side of Pluto is always faces the same side of Charon.

Imagine you have a bucket half full of water. Can you tilt a bucket completely sideways without spilling a drop?

Sure thing! You can swing it by the handle, and even though it’s upside down at one point, the water stays put.

What’s keeping the water inside the bucket?

Before we answer this, imagine you are a passenger in a car, and the driver is late for an appointment.

They take a turn a little too fast, and you forgot to fasten your seat belts. The car makes a sharp left turn. Which way would you move in the car if they took this turn too fast? Exactly – you’d go sliding to

the right. So, who pushed you?

No one! Your body wanted to continue in a straight line, but

the car is turning, so the right side car door keeps pushing you

to turn you in a curve – into the left turn. The car door keeps

moving in your way, turning you into a circle. The car door

pushing on you is called centripetal force. Centripetal means

“center-seeking.” It’s the force that points toward the center of

the circle you’re moving on. When you swing the bucket

around your head, the bottom of the bucket is making the

water turn in a circle and not fly away. Your arm is pulling on

the handle of the bucket, keeping it turning in a circle and not

letting it fly away. That’s centriprtal force.

Think of it this way: If I throw a ball in outer space, does it go in a straight line or does it wiggle all over the place? Straight line, right? Centripetal force is the force needed to keep an object following a curved path.

Remember how objects will travel in a straight line unless they bump into something or have another

force acting on them, such as gravity, drag force, and so forth? Well, to keep the bucket of water

swinging in a curved arc, the centripetal force can be felt in the tension experienced by the handle (or

your arm, in our case). Swinging an object around on a string will cause the rope to undergo tension

(centripetal force), and if your rope isn’t strong enough, it will snap and break, sending the mass flying

off in a tangent (straight) line until gravity and drag force pull the object to a stop. This force is

proportional to the square of the speed - the faster you swing the object, the higher the force.


Exercises

1. How is spinning the container like Pluto and Charon?

2. What would happen if we cut the string while you are spinning? Which way would the container go?

3. What happens if we triple the size of your container and what’s inside of it?


Answers to Exercises: Binary Planetary Systems

1. How is spinning the container like Pluto and Charon? (You are always facing the same side of the container, just like Pluto and Charon are always facing the same sides of each other.)

2. What would happen if we cut the string while you are spinning? Which way would the container go? (In a straight line tangent to the curve at the moment we cut the string.)

3. What happens if we triple the size of your container and what’s inside of it? (It takes more energy to

swing a larger load around. For one object to orbit another, they must have strong gravitational

attraction to move that much mass around.)


WHAT’S UP IN THE SKY?

Today you get to learn how to read an astronomical chart to find out when the Sun sets, when twilight

ends, which planets are visible, when the next full moon occurs, and much more. This is an excellent way

to impress your friends! The patterns of stars and planets stay the same, although they appear to move

across the sky nightly, and different stars and planets can be seen in different seasons.


What’s in the Sky? Overview: Today you get to learn how to read an astronomical chart to find out when the Sun sets, when twilightends, which planets are visible, when the next full moon occurs, and much more. This is an excellent way to impress your friends.

What to Learn: The patterns of stars and planets stay the same, although they appear to move across the skynightly, and different stars and planets can be seen in different seasons.

Materials

Printout of Stargazer’s Almanac

Pencil

Tape and scissors (optional)

Ruler

Experiment

1. If your chart comes on two pages, you’ll need to cut the borders off at the top and bottom and tape them together so they fit perfectly.

2. Use your ruler as a straight edge to help locate items as you read the chart.

3. Print out copies of the almanac by clicking the image of the Skygazer’s Almanac. You can print it full-size on

two pages, or size it to fit onto a single page. Since there’s a ton of information on it, it’s best read over two

pages. This is an expired calendar to practice with.

4. First, note the “hourglass” shape of the chart. Do you see how it’s skinnier in the middle and wider near the

ends? Since it’s an astronomical chart that shows what’s up in the sky at night, the nights are shorter during

the summer months, so the number of hours the stars are visible is a lot less than during the winter. You’ll

find the hours of the night printed across the top and bottom of the chart (find it now) and the months and

days of the year printed on the right and left side.

5. Can you find the summer solstice on June 20? Use your finger and start on the left side between June 17 and June 24. The 20th is between those two dates somewhere. Here’s how you tell exactly…

6. Look at the entire chart – do you see the little dots that make up little squares all over the chart, like a grid? Each

dot in the vertical direction represents one day. There are eight dots on the vertical side of the box.

7. Let’s say you want to find out what time Neptune rises on June 17. Go back to June 17, which has its own little set of dots. Follow the dots with your finger until you hit the line that says Neptune Rises. Stop and

trace it up vertically to the top scale to read just after 11 p.m.

8. Look again at the dot boxes. Each horizontal dot is 5 minutes apart, and every six dots there is a vertical

line representing the half-hour. The line crosses between the second and third dot, so if you lived in a place

where you can clearly see the eastern horizon and looked out at 11:07, you’d see Neptune just rising. Since


Uranus and Neptune are so far away, though, you’d need a telescope to see them. So let’s try something you

can find with your naked eye.

9. Look at Oct 21. What time does Saturn set? (5:30p.m.).

10. What other two planets set right afterward? (Mercury at 6:03 p.m. and Mars sets at 7:12 p.m.).

11. When does Jupiter rise? (7:32 p.m.).

12. What is Neptune doing that night of Oct. 21? (Neptune transits, or is directly overhead, at 8:07 p.m. and sets at 1:30 a.m.)

13. What other interesting things happen on Oct. 21? (Betelgeuse, one of the bright stars in the constellation

Orion, rises at 9:23 p.m. Sirius, the dog star, rises at 11:06 p.m. The Pleiades, also known as the Seven

Sisters, are overhead at 1:42 a.m.)

14. Let’s find out when the Moon rises on Oct. 21. You’ll find a half circle representing the Moon centered on 11:05 p.m. Which phase is the Moon at? First or third quarter? (First. You can tell if you look at the next couple of days to see if the Moon waxes or wanes. Large circles indicate one of the four main phases of the Moon.)

15. When does the Sun rise and set for Oct. 21? First, find the nearest vertical set of dots and read the time (5:30 p.m.). Now subtract out the 5-minute dots until you get to the edge. You should read three dots plus a little extra, which we estimate to be 17 minutes. Sunset is at 5:13 p.m. on Oct 21.

16. Note the fuzzy, lighter areas on both sides of the hourglass. That represents the twilight time when it’s not

quite dark, but it’s not daylight either. There’s a thin dashed line that runs up and down the vertical,

following the curve of the hourglass offset by about an hour and 35 minutes. That’s the official time that

twilight ends and the night begins.

17. Can you find a meteor shower? Look for a starburst symbol and find the date right in the center. Those are

the peak times to view the shower, and it’s usually in the wee morning hours. The very best meteor

showers are when there’s also a new Moon nearby.

18. Notice how Mercury and Venus stay close by the edges of the twilight. You’ll find a half-circle symbol

representing the day that they are furthest from the Sun as viewed from the Earth, which is the best date to

view it. For Venus, the * indicates the day that it’s the brightest.

19. What do you think the open circle means at sunset on May 20? (New Moon)

20. Students that spot the “Sun slow” or “Sun fast” marks on the chart always ask about it. It’s actually rather

complicated to explain, but here’s the best way to think about it. Imagine that the vertical timeline running

down the center means noon, not midnight. Do you see a second line weaving back and forth across the

noon line throughout the year? That’s the line that shows the when the Sun crosses the meridian. On Feb 5,

the Sun crosses that meridian at 12:14, so it’s “running slow,” because it “should have” crossed the

meridian at noon. This small variation is due to the axis tilt of the Earth. Note that it never gets much more

than 15 minutes fast or slow. The wavy line that represents this effect is called the Equation of Time. We’ll

be using that later when we make our own sundials and have to correct for the Sun not being where it’s

supposed to be.


21. Look at Mars and Saturn both setting around the same time on Aug. 14. When two event lines cross, you’ll find

nearby an open circle with a line coming from the top right side, accompanied by a set of arrows pointing

toward each other. This means conjunction, and is a time when you can see two objects at once.

Usually the symbol isn’t right at the intersection, because one of the objects is rising or setting and isn’t

clearly visible. On Aug. 14, you’ll want to view them a little before they set, so the symbol is moved to a time

where you can see them both more clearly.

22. Important to note: If your area uses daylight savings time, you’ll need to add one hour to the times shown on the chart.

23. Time corrections for advanced students: This chart was made for folks living on the 40onorth latitude and90o west longitude lines (which is Peoria, Ill.).

a. If you live near the standardized longitudes for Eastern Time (75o), Central (90o), Mountain (105o)

or Pacific (120o), then you don’t have to correct the chart times you read. However, if you live a

little west or east of these standardized locations, you need a correction, which looks like this:

i. For every degree west, add four minutes to the time you read off the chart.

ii. For every degree east, subtract four minutes from the time.

iii. For example, if you lived in Washington, D.C. (which is 77o longitude), note that this is 2o west of the Eastern Time, so you’d add 8 minutes to the time you read off the chart.

iv. Memorize your particular adjustment and always use it.

b. If your latitude isn’t 40o north, then you need to adjust the rise and set times like this:

i. If you live north of 40o, then the object you are viewing will be in the sky for longer than the chart shows, as it will rise earlier and set later.

ii. If you live south of 40o, then the object you are viewing will be in the sky for less time than the chart shows, as it will rise later and set earlier.

iii. The easiest way to calculate this is to note what time an object should rise, and then watch

to see when it actually appears against a level horizon. This is your correction for your

location.

4. Complete the data table.


“What’s in the Sky?” Data Table

Question Answer (date and time)

What time does Venus set on April 22?

When does Mars set on August 12?

When is the full Moon in March?

When is the best date and time to view both Jupiter and

Saturn?

When is the best meteor shower for the entire year?

Which day is the longest?

When do two planets rise at the same time?

If this calendar was for this year at your exact location, what

would you be looking forward to tonight?

Reading

This is one of the finest charts I’ve ever used as an astronomer, as it has so much information all in one place.

You’ll find the rise and set times for all eight planets, peak times for annual meteor showers, moon phases, sunrise

and set times, and it gives an overall picture of what the evening looks like over the entire year. Kids can clearly

see the planetary movement patterns and quickly find what they need each night. I keep one of these posted right

by the door for everyone to view all year long.


Exercises

1. Is Mercury visible during the entire year?

2. In general, when and where should you look for Venus?

3. When is the best time to view a meteor shower?

4. Which date has the most planets visible in the sky?


Answers to Exercises: What’s in the Sky?

“What’s in the Sky?” Data Table

Question Answer (date and/or time)

What time does Venus set on April 22? 10:35 p.m.

When does Mars set on August 12? 9:30 p.m.

When is the full Moon in March? March 8

When is the best date and time to view both Jupiter and Saturn? (answers vary, but Sept. 9 is a choice)

When is the best meteor shower for the entire year?

(answers vary, but Lyrids and Leonids are

great choices with nearly no moon)

Which day is the longest? Dec. 21

When do two planets rise at the same time? 4:30 a.m. on Nov. 27

If this calendar was for this year at your exact location, what

(answers vary)

would you be looking forward to tonight?

1. Is Mercury visible during the entire year? (No, only for a couple of months.)

2. In general, when and where should you look for Venus? (Near the Eastern or Western sky during twilight during certain months of the year, because it’s always rising or setting, never transiting.)

3. When is the best time to view a meteor shower? (Look for a starburst symbol that is close to a new moon symbol. The skies will be dark enough to view the meteors.)

4. Which date has the most planets visible in the sky? (Feb. 12 has all 7 planets visible sometime during the night, although Nov. 11 is a better night to view, since Mercury and Neptune won’t be lost in the sunset.)


BUILD A SOLAR SYSTEM

What would happen if our solar system had three suns? Or the Earth had two moons? You can find out all these and more with this lesson on orbital mechanics. Instead of waiting until you hit college, we thought we’d throw some university-level physics at you… without the hard college-level math. To get you experienced with the force of gravity without getting lost in the math, there’s an excellent computer program that allows you to see how multi-object systems interact. Most textbooks are limited to the interaction between a very large object, like the Earth, and much smaller objects that are very close to it, like the Moon. This seriously cuts out most of the interesting solar systems that are out there in the real universe. The University of Colorado at Boulder designed a great system to do the hard math for you. Don’t be fooled by the simplistic appearance – the physics behind the simulation is rock-solid… meaning that the results you get are exactly what scientists would predict to happen.


Build Your Own Solar System Overview: What would happen if our solar system had three Suns? Or the Earth had two moons? You can find outall these and more with this lesson on orbital mechanics. Instead of waiting until you hit college, we thought

we’d throw some university-level physics at you… without the hard math.

What to Learn Key concepts about gravity:

a. Gravity is a force that attracts things to one another.

b. All bodies (objects) have a gravitational field.

c. The larger a body is, the greater the strength of the gravitational field.

d. Bodies must be very, very large before they exert any noticeable gravitational field.

e. Gravity accelerates all things equally. Which means all things speed up the same amount as they fall.

f. Gravity does not care what size things are or whether things are moving. All things are accelerated toward the Earth at the same rate of speed.

g. Gravity does pull on things differently. Gravity is pulling greater on objects that weigh more.

h. Weight is a measure of how much gravity is pulling on an object.

i. Mass is a measure of how much matter (how many atoms) make up an object.

Materials

Access to a computer with Internet

Ruler

Experiment

About the Concept of Gravity

1. Even though we deal with gravity on a constant basis, there are several misconceptions about it. Let’s get to an experiment right away and I’ll show you what I mean.

2. When you drop a golf ball and a ping pong ball from the same height, what happens?

3. What you should see is that both objects hit the ground at the same time! Gravity accelerates both items

equally and they hit the ground at the same time. Any two objects will do this, a brick and a Buick, a

flower and a fish, a kumquat and a cow!

4. But what if you drop a feather and a ball at the same time? There is one thing that will change the results

and that is air resistance. The bigger, lighter and fluffier something is, the more air resistance can affect it

and so it will fall more slowly. Air resistance is a type of friction which we will be talking about later. In

fact, if you removed air resistance, a feather and a flounder would hit the ground at the same time!!!


5. Where can you remove air resistance? The moon!!! One of the Apollo missions actually did this (well, they didn’t use a flounder, they used a hammer). An astronaut dropped a feather and a hammer at the same time and indeed, both fell at the same rate of speed and hit the surface of the moon at the same time.

6. Which will hit the ground first, if dropped from the same height, a bowling ball or a tennis ball? Most

people will say the bowling ball. In fact, if you asked yourself that question 5 minutes ago, would you have

gotten it right? It’s conventional wisdom to think that the heavier object falls faster.

7. Unfortunately, conventional wisdom isn’t always right. Gravity accelerates all things equally. In other words, gravity makes all things speed up or slow down at the same rate.

8. This is a great example of why the scientific method (more on this later) is such a cool thing. Many, many

years ago, there was a man of great knowledge and wisdom named Aristotle. Whatever he said, most

people believed to be true. The trouble was, he didn’t test everything that he said. One of his statements

was that objects with greater weight fall faster than objects with less weight. Everyone believed that this

was true. Hundreds of years later, Galileo came along and said “Ya know...that doesn’t seem to work that

way. I’m going to test it” The story goes that Galileo grabbed a melon and an orange and went to the top of

the Leaning Tower of Pisa. He said, “Look out below!” and dropped them! By doing that, he showed that

objects fall at the same rate of speed no matter what their size. It is true that it was Galileo who “proved”

that gravity accelerates all things equally no matter what their weight, but there is no real evidence that

he actually used the Leaning Tower of Pisa to do it.

9. Key concepts about Gravity:

a. Gravity is a force that attracts things to one another.

b. All bodies (objects) have a gravitational field.

c. The larger a body is, the greater the strength of the gravitational field.

d. Bodies must be very, very large before they exert any noticeable gravitational field.

e. Gravity accelerates all things equally. Which means all things speed up the same amount as they fall.

f. Gravity does not care what size things are or whether things are moving. All things are accelerated toward the Earth at the same rate of speed.

g. Gravity does pull on things differently. Gravity is pulling greater on objects that weigh more.

h. Weight is a measure of how much gravity is pulling on an object.

i. Mass is a measure of how much matter (how many atoms) make up an object.

About the Computer Simulation

1. To get you experienced with the force of gravity without getting lost in the math, there’s an excellent

computer program that allows you to see how multi-object systems interact. Most textbooks are limited

to the interaction between a very large object, like the Earth, and much smaller objects that are very close

to it, like the Moon. This seriously cuts out most of the interesting solar systems that are out there in the

real universe.


2. The University of Colorado at Boulder designed a great system to do the hard math for you. Don’t be fooled by the simplistic appearance – the physics behind the simulation are rock-solid… meaning that the results you get are exactly what scientists would predict to happen.

3. Go to the My Solar System simulation on the PhET website and carefully follow the instructions for

each activity. Answer the questions and record your results before going on to the next activity. Visit:

http://phet.colorado.edu/sims/my-solar-system/my-solar-system.swf

4. Once the program opens, hit start. You’ll see the purple Earth orbit around the yellow Sun. Do you

notice how the Earth also causes the Sun to follow a tiny orbit? That’s because the Earth pulls on the Sun just as the Sun pulls on the Earth.

5. Press stop. Notice the “V?” That stands for direction and speed, as in 55 mph north. It gives how fast you

are going as well as the direction you’re going. Or in this case, the planet. Notice near the bottom that

you can change the mass of the object. Increase the mass so that it’s larger than the Sun. Press start.

6. Reset, and change the purple object (Earth) to be the size of the Moon (make it 1). Did you notice a change in the orbit path?

7. Change the purple mass back to 10, and increase the speed to a larger number. What happened?

8. The Earth is at a very special mass and speed, isn’t it?

9. Reset and make your speed 200. Did it stay in orbit?

10. Add a third and fourth object by pulling down the menu on the upper right. Select “Sun, planet, and moon” and hit “Start”.

11. What happens if you uncheck “show trace”? (You’ll only see the objects themselves orbiting, not the path they take.)

12. . What happens if you uncheck “system centered”? (The system will eventually wander off the screen as the entire system has acceleration.)


Play with the program for a bit, changing the location distance the objects are apart, the speed and direction they initially start out at, and their masses.

What does the yellow object represent?________________________________________ What is the mass of the yellow object? ________________________________________

(Note: No units are given, so no units are necessary.)

What does the purple object represent? _______________________________________

What is the mass of the purple object? ________________________________________

What does the red arrow represent? ________________________________________

Complete the data table. Notice at the end that you will predict the necessary mass, velocity, and distance from the Sun of a planet in order for this planet to make a circular orbit around a Sun.

Reading

In 1666, Isaac Newton did his early work on his Three Laws of Motion. To this day, those laws still hold true.

There has been some allowances for really big things (like the cosmos) and for really small things (like the atom).

Other than that, Newton’s Laws are pretty much dead on.

Newton’s Laws are all scientists and engineers used to get the first man to the moon. They are an amazingly

powerful and wonderful area of physics. I like them because evidence of them is everywhere. If something moves

or can be moved, it follows Newton’s Laws. You can’t sit in a car, walk down the road, drink a glass of milk, or kick

a ball without using Newton’s Laws. I also like them because they are relatively easy to understand and yet open

up worlds of answers and questions. They are truly a foundation for understanding the world around you.

Whenever I teach a class about gravity, I’ll drop something (usually something large). After the heads whip around, I ask the hard question: “Why did it fall?” You already know the answer – gravity.

But why? Why does gravity pull things down, not up? And when did people first start noticing that we stick to the surface of the planet and not float up into the sky?

No one can tell you why gravity is… that’s just the way the universe is wired. Gravitation is a natural thing that happens when you have mass. Galileo was actually one of the first people to do science experiments on gravity.


Build Your Own Solar System Data Table 1

Use the original preset for all values for a Sun and Planet,

except change the mass of body 2 (purple object) as shown below:

Exercises: (Note that the exercise questions are below each data table)

1. What effect does changing the mass of orbiting planet have on the diameter of the orbit?

Mass Body 2

Diameter Of Orbit

(measure with ruler)

1000

100

10

1

0.1

0.01

0.001

0.0001




except change the mass of body 2 (purple object) and velocity as shown below:

Mass of Body 2 Velocity of Body 2 Describe what happened…?

0.1 y velocity = 130




0.1 y velocity = 80

0.1 y velocity = 40

0.1 y velocity = 20

0.1 y velocity = 0

2. What effect does changing the speed have on a planet’s orbit?




except change the mass of body 2 to 50 and the x-distance of body 2 as shown below:

x distance for body 2

Diameter Of Orbits ( measure of ruler )

30

60

90

120

150

180

210

240

3. What happens to the planet's orbit when you increase the initial distance between the planet and the Sun?



Use the original preset for a Binary Star and Planet. Change only the masses and record your observations below.

Mass of Mass of Mass of Is the orbit stable?

Body 1 Body 2 Body 3

5. Find the mass values needed for a stable orbit. Circle the values on the table that make a stable orbit.



Use the original preset for Ellipses. Change only the masses and record your observations below.

Mass of Mass of Mass of Mass of What happened?

Body 1 Body 2 Body 3 Body 4

250 10 1 0.1

5. Why don’t a feather and a brick hit the ground at the same time?


Answers to Exercises: Build Your Own Solar System

1. What effect does changing the mass of orbiting planet have on the diameter of the orbit?

2. What effect does changing the speed have on a planet’s orbit?

3. What happens to the planet's orbit when you increase the initial distance between the planet and the Sun?

4. Find the mass values needed for a stable orbit. Circle the values on the table that make a stable orbit.

5. Why don’t a feather and a brick hit the ground at the same time? (They do…if you’re on the moon! On Earth, the friction between the air and the feather causes the feather to slow down and the brick to win the race.)


RETROGRADE MOTION

If you watch the moon, you’d notice that it rises in the east and sets in the west. This direction is called ‘prograde motion’. The stars, sun, and moon all follow the same prograde motion, meaning that they all move across the sky in the same direction. However, at certain times of the orbit, certain planets move in ‘retrograde motion’, the opposite way. Mars, Venus, and Mercury all have retrograde motion that have been recorded for as long as we’ve had something to write with. While most of the time, they spend their time in the ‘prograde’ direction, you’ll find that sometimes they stop, go backwards, stop, then go forward again, all over the course of several days to weeks. Satellite orbits can either be circular or elliptical. For circular orbits, the satellite is moving with a constant speed at the same height above the earth’s surface for the entire trip, because it’s falling toward the earth at exactly the same rate that the earth is curving away from it (about 5 meters down every 8,000 meters horizontally traveled). Since the force of gravity always points inward (because the object is on a circular path), it’s independent of the constant velocity vector (which is tangent to the circular path), and the angle between the force and the motion direction is always at 90 degrees, so work is not done by the force of gravity on the satellite. However, for an elliptical orbit, there is a component of force in the direction of motion, which means that the force does do work on the satellite, so the force can either speed up or slow down the satellite. For example, when a satellite moves toward the earth, the force is in the same direction as the motion so it increases the speed of the satellite. Negative work indicates that the satellite is slowing down. This means that the speed of the satellite is constantly changing.


Retrograde Motion

Overview: Three planets, Mars, Mercury, and Venus, appear to move backward in the sky when tracked night afternight. This motion is called “Retrograde Motion” and has baffled scientists for years.

What to Learn: From a top view of the solar system, the planets appear to move around the Sun in an

orderlyfashion. The real chaos comes in when you place yourself on one of these planets and try to watch the path

that the others take while you’re orbiting the Sun. It’s predictable chaos, though, with enough math and physics

under your belt (like in college). Today you’re just going to get a sneak peek at the wild world of orbital mechanics.

Materials

Pencil

Ruler

Experiment

1. Look at the diagram on the next page. The tiny center circle (without any dots) is the Sun. The inner circle is the Earth’s orbit, and the other circle is the orbit of Mars. The dots show where Mars and the Earth are each month. The dashed line is the sky we’d see on Earth.

2. I’ve already drawn a line with my ruler connecting the two January dots. (I know it also went through February, but that’s because it just happened to be there.)

3. Take your ruler and connect the two dots for February. Make sure to extend your lines a little past the sky before labeling the end of the line with a 2.

4. Do this for each month, connecting the dots starting with the inner Earth circle month to the corresponding Mars circle month. The March months should have a 3 label at the end.

5. If you find that your lines cross, make the lines a little longer and make the dots further away so you can tell which number goes with which line.

6. Now for the fun part: Play “connect the dots” with the numbered dots in the sky. Start with the 1, and carefully connect your dots in order. This line is the path that Mars will follow when you look at Mars from Earth.


The Sky

Mars Orbit

1

Earth Orbit


Reading

If you watch the moon, you’d notice that it rises in the east and sets in the west. This direction is called

“prograde motion.” The stars, Sun, and moon all follow the same prograde motion, meaning that they all move across the sky in the same direction.

However, at certain times of the orbit, certain planets move in “retrograde motion,” the opposite way.

Mars, Venus, and Mercury all have retrograde motion that have been recorded for as long as we’ve had

something to write with. While most of the time, they spend their time in the “prograde” direction, you’ll

find that sometimes they stop, go backward, stop, and then go forward again, all over the course of

several days to weeks.

It’s like going down a racetrack on the inside curve. You pass the outside car quickly, and from your point of view, they seem to be moving backward as you pass them.

Here are videos I created that show you what this would look like if you tracked their position in the sky each night for a year or two.

Mercury and Venus Retrograde Motion

This is a video that shows the retrograde motion of Venus and Mercury over the course of several years.

Venus is the dot that stays centered throughout the video (Mercury is the one that swings around

rapidly), and the bright dot is the Sun. Note how sometimes the trace lines zigzag, and other times they

loop. Mercury and Venus never get far from the Sun from Earth’s point of view, which is why you’ll only

see Mercury in the early dawn or early evening.

Retrograde Motion of Mars

You’ve probably heard of epicycles people used to use to help explain the retrograde motion of Mars.

Have you ever wondered what the fuss was all about? Here’s a video that traces out the path Mars takes

over the course of several years. Do you see our Moon zipping by? The planets, Sun, and Moon all travel

along line called the ”ecliptic,” as they all are in about the same plane.

Several planets found outside our solar system (called extrasolar planets) have backward orbits.

This isn’t retrograde motion, just plain old backward… something we’ve never seen before in our

search for extrasolar planets!

Mars retrogrades for 72 days every 25.6 months, Jupiter for 121 days every 13.1 months, Saturn for 138 days every 12.4 months, Uranus for 151 days every 12.15 months, and Neptune for 158 days every 12.07 months.


Exercises

1. During which of the months does Mars appear to move in retrograde?

2. Why does Mars appear to move backward?

3. Which planets have retrograde motion?

Answers to Exercises: Retrograde Motion

1. During which months does Mars move in retrograde? (Between April and July)

2. Why does Mars appear to move backward? (As the Earth passes Mars more quickly, Mars appears to slow down, stop, and reverse direction.)

3. Which planets have retrograde motion? (All planets.)


POTENTIAL ENERGY IN THE STARS

In our physics problems so far, we’ve kept the objects close to the earth so that the acceleration due to gravity gremainsconstant, and we defined the potential energy of an object on the surface of the earth to be zero. What if we look up and see three stars in a system and want to find out the gravitational potential energy of the system?

Globular clusters have a HUGE amount of gravitational potential energy.


ESCAPE SPEED

Let’s determine the gravitational potential energy of the earth-moon system as well as the speed needed to escape the Earth’s gravitational pull:


CONSERVATION OF MECHANICAL ENERGY

Here’s how you put it all together and figure out the total energy of the system. This is useful when

you’re trying to figure out something that you can’t otherwise solve for… let me show you with a set of

videos here. Remember, for satellites the only force we have on the object is due to gravity, so the

external work force term always goes to zero like this:


GEOSYNCHRONOUS ORBIT

A geosynchronous orbit is an orbit that a satellite has when viewed from the earth, looks like the

satellite is stationary.


GRAVITATIONAL POTENTIAL ENERGY

How do you find the gravitational potential energy between two objects that are really far apart, like two

stars or two galaxies? Here’s how:


ROCKETSHIP RACES

Imagine you and I are racing rocketships in orbit around the Earth. I can slow down and still beat you

around the Earth. Want to see how?


EQUIVALENCE PRINCIPLE

Einstein once said: “I was sitting in a chair in the patent office at Bern when all of the sudden a thought occurred to me: If a person falls freely, he will not feel his own weight. I was startled. This simple thought made a deep impression on me. It impelled me toward a theory of gravitation.” This led Einstein to develop his general theory of relativity, which interprets gravity not as a force but as the curvature of space and time. This topic is out of the scope for our lesson here, but you can explore more about it in this lesson. The fundamental principle for relativity is the principle of equivalence, which says that if you were locked up in a box, you wouldn’t tell the difference between being in a gravitational field and accelerating (with an acceleration value equal to g) in a rocket. The same thing is also true if you were either locked in a box, floating in outer space or in an elevator shaft experiencing free-fall. Any experiments you could do in either of those cases wouldn’t be able to tell you what was really happening outside your box. The way a ball drops is exactly the same in either case, and you would not be able to tell if you were falling in an elevator shaft or drifting in space. Have you ever seen an icicle? They usually grown downward toward the center of the earth (the direction of free-fall). But icicles on a car wheel grow radially outward because the wheel spins and flings the water toward the outside of the wheel where it freezes into spikes. The icicles can’t tell if the wheel is rotating and that’s why they grow radially, or it’s at rest at the gravitational field is in the radial direction! Here’s a questions for you: these two astronauts (below) are inside the space station, which is currently in orbit around the earth. Which astronaut is upside down? Can you tell? The principle of equivalence has some consequences! Navigation systems for ships, airplanes, missiles and submarines rely on acceleration information to calculate their velocity and position. However, the instruments that measure acceleration also react with unexpected variations in the earth’s gravitational field, and there’s no direct way to separate these two effect to avoid errors. Highlights for Kepler’s Laws:

The Law of Orbits: All planets move in elliptical orbits, with the sun at one focus.

The Law of Areas: A line joining the planet to the sun sweeps out equal areas in equal times.

The Law of Periods: The square of the period of any planet is proportional tot he cube of the semi-major axis of its orbit.

Yay! You completed this set of lessons on circular motion! Now it’s time for you to work your own physics problems!


HOMEWORK PROBLEMS WITH SOLUTIONS

On the following pages is the homework assignment for this unit. When you’ve completed all the videos from this unit, turn to the next page for the homework assignment. Do your best to work through as many problems as you can. When you finish, grade your own assignment so you can see how much you’ve learned and feel confident and proud of your achievement! If there are any holes in your understanding, go back and watch the videos again to make sure you’re comfortable with the content before moving onto the next unit. Don’t worry too much about mistakes at this point. Just work through the problems again and be totally amazed at how much you’re learning. If you’re scoring or keeping a grade-type of record for homework assignments, here’s my personal philosophy on using such a scoring mechanism for a course like this: It’s more advantageous to assign a “pass” or “incomplete” score to yourself when scoring your homework assignment instead of a grade or “percent correct” score (like a 85%, or B) simply because students learn faster and more effectively when they build on their successes instead of focusing on their failures. While working through the course, ask a friend or parent to point to three questions you solved correctly and ask you why or how you solved it. Any problems you didn’t solve correctly simply mean that you’ll need to go back and work on them until you feel confident you could handle them when they pop up again in the future.

Advanced Physics Uniform Circular Motion

Student Worksheet for Uniform Circular Motion

After you’ve worked through the sample problems in the videos, you can work out the problems below to practice doing this yourself. Answers are given on the last page.

Friction:

Fk = μkN

Fs = μsN

Centripetal Force:

a F = m = Rmv2

Law of Gravitation:

F = r2Gm m1 2

Free-fall acceleration:

go = r2GM

Gravitational Potential Energy:

)U = − ( r122

Gm m1 2 + r132

Gm m1 3 + r232

Gm m2 3 + …

Escape Speed:

v = √ R2GM

Law of Periods:

) r3T2

= (4π2

GM

Energy in Planetary Motion: E = K + U

and K U = − rGMm = 2r

GMm

© 2014 Supercharged Science www.ScienceLearningSpace.com Page 1


Practice Problems:

1. A car travels around a corner on an unbanked track. The radius of the track is 12 meters and the car is traveling at a velocity of 5 meters per second. Find the angle that a strap hanging from the inside of the car will make as the car travels around the corner.

2. A student stands on a merry-go-round of radius 4.3 m. The merry-go-round revolves once every 30 seconds. Find the speed of the child on the outer rim of the merry-go-round. What must be the coefficient of static friction between the boys shoes and the merry-go-round for him to stay on the merry-go-round?

3. A bicyclist rides in a circle with a radius of 20 m at a speed of 5 m/s. If the combined mass of the bike and the rider is 90 kg, what is the force of friction exerted on the bike by the road? What is the total force exerted by the road on the bike?



4. A child of mass 73 kg is on a Ferris wheel. At the top of the Ferris wheel the child has an apparent weight of 637N. What is the child’s apparent weight at the lowest point? Find the apparent weight of the student at the highest point if the speed of the Ferris wheel was tripled.

5. A car driver takes a turn of radius 10 m at a speed of 25 m/s. How many g’s of acceleration will the the driver experience?

6. An 11kN car rounds an unbanked curve with a radius of 70 m at a speed of 15 m/s. Find the force of friction of the road on the tires to keep the car on its path. If the coefficient of friction between the tires and the road is 0.33, will the car slip out of its turn?



7. Which takes more energy: Putting a satellite up to 1500 km or putting a satellite in orbit that is already at 1500 km? What about at 3000 km and 4500 km instead of 1500km? The Earth’s radius is 6000 km.

8. An asteroid with mass 1.2x10-4 times the mass of the Earth revolves in a circular orbit around the sun. The asteroid’s orbit is 1.5 times that of Earth’s. Find the period the asteroid’s revolution in Earth years. What is the ratio between the kinetic energy of the asteroid and Earth?

9. Assuming the Earth travels in a circular orbit, how far does the Earth travel in 70 seconds?



10. Phobos orbits the planet Mars at a radius of 9.3x106 m with a period of 8 hours. Find the mass of Mars.

11. If a satellite is placed into orbit, around the Earth, at a radius of three quarters of the Moon’s orbit radius, what is the satellite’s period of revolution in lunar months?

12. A spaceship is 77,850 light years from the center of a galaxy. The mass of the galaxy 1.4x1011 times the mass of our sun. Assume the galaxy acts as a spherical distribution so you can use of the law of gravitation. What minimum speed must the spaceship have to escape the galaxy?



13. What is the escape velocity of a planet of radius 700 km with gravitational acceleration one half of Earth’s gravitational acceleration? How high will a particle go if it is launched up with an initial speed of 998 m/s on this planet? What speed will the object reach if it is dropped from a height of 2000 km (2300 km from the center of the planet?)

14. In a double star system, two stars orbit about their shared center of mass. Both stars share the same mass of 3.0x1030 kg and are a distance of 2.0x1011 m from each other. What is their angular speed? If a meteorite travels through the center of mass between the two planets what must its speed be to escape the gravitational field of the two stars?

15. Find the gravity on the surface of the moon. What is the period of a pendulum on the moon, if the pendulum normally has a period of three seconds on Earth?



16. A person weighs 600 N at sea level on Earth. If you ride an elevator up to 500 meters, what will be your new apparent weight?

17. If you draw a line from the center of Earth to the center of the Sun. Where must a particle be on that line in order for the gravitational field of the Earth and the Sun the cancel eachother out?

18. What is the ratio between Earth’s acceleration towards the sun when the moon is between the Earth and the sun compared to when the Earth is between the moon and the sun?


AdvancedPhysics UniformCircularMotion

©2014SuperchargedScience www.ScienceLearningSpace.com Page1

StudentWorksheetforUniformCircularMotionAfteryou’veworkedthroughthesampleproblemsinthevideos,youcanworkouttheproblemsbelowtopracticedoingthisyourself.Answersaregivenonthelastpage.

Friction:

Fk = μkN

Fs = μsN

Centripetal Force:

F = ma = mRv2

Law of Gravitation:

F = Gmr12m2

Free-fall acceleration:

go = GrM2

Gravitational Potential Energy:

U = − (Gmr112m2 2 + Gmr113m2 3 + Gmr223m2 3 + …)

Escape Speed:

v = √2GRM

LawofPeriods:

Tr32 = (G4πM2)

EnergyinPlanetaryMotion:E=K+U

U = − GMr m and K = G2Mrm

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circlewitha90kg,whatytheroado

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9

pm ��

nm ��

em ��

c � �

e ��

k pe� � � �

G ��

g �

�

W

�

�

� m�

�

q � � � � � � �

q

q

q �

MECHANICS ELECTRICITY

0x x xa tÃ Ã� �

20 0

12xx x t aÃ� � � xt

�2 20 2x x xa x xÃ Ã� � � 0

netFFa

m� ��

m

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p mv��

p F tD D��

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cosE W F d Fd qD � � ��

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20 0

12

tq q w a� � � t

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net

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sinr F rFt �� q

L Iw�

L ttD � D

212

K Iw�

sF k x��

2

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mV

r �

a = acceleration A = amplitude d = distance E = energy f = frequency F = force I = rotational inertia K = kinetic energy k = spring constant L = angular momentum � = lengthm = mass P = power p = momentum r = radius or separation T = period t = time U = potential energy V = volume v = speed W = work done on a system x = position y = height a = angular acceleration

m = coefficient of friction q = angle r = density t = torque w = angular speed

gU mg yD � D

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pw

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p�

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p� �

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m mF G

r�

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q qF k

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si

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WAVES

vf

l � f = frequency v = speed l = wavelength

GEOMETRY AND TRIGONOMETRY

Rectangle A bh�

Triangle 12

A bh�

Circle 2A p�

2C p�r

r

Rectangular solid V wh� �

Cylinder

V rp� �2

2 2S r rp p� �� 2

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3V rp� 4

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2 2 2c a b� �

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q �

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q �

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CONSTANTS AND CONVERSION FACTORS

Proton mass, 271.67 10 kgpm ��

Neutron mass, 271.67 10 kgnm ��

Electron mass, 319.11 10 kgem ��

Avogadro’s number, 23 -10 6.02 10 molN � �

Universal gas constant, 8.31 J (mol K)R � �

Boltzmann’s constant, 231.38 10 J KBk ��

Electron charge magnitude, 191.60 10 Ce ��

1 electron volt, 191 eV 1.60 10 J�� Speed of light, 83.00 10 m sc � �

Universal gravitational constant,

11 3 26.67 10 m kg sG ��

Acceleration due to gravityat Earth’s surface,

29.8 m sg �

1 unified atomic mass unit, 27 21 u 1.66 10 kg 931 MeV c�� Planck’s constant, 34 156.63 10 J s 4.14 10 eV sh ��

25 31.99 10 J m 1.24 10 eV nmhc ��

Vacuum permittivity, 12 2 20 8.85 10 C N me ��

Coulomb’s law constant, 9 201 4 9.0 10 N m Ck pe� � � �

AVacuum permeability, 70 4 10 (T m)m p ��

Magnetic constant, 70 4 1 10 (T m)k m p ��

5 1 atmosphere pressure, 5 21 atm 1.0 10 N m 1.0 10 Pa� � � �

UNIT SYMBOLS

meter, m kilogram, kgsecond, sampere, Akelvin, K

mole, mol hertz, Hz

newton, Npascal, Pajoule, J

watt, W coulomb, C

volt, Vohm,

henry, H

farad, F tesla, T

degree Celsius, C� W electron volt, eV

2

A

�

�

PREFIXES Factor Prefix Symbol

1012 tera T

109 giga G

106 mega M

103 kilo k

10�2 centi c

10�3 milli m

10�6 micro m

10�9 nano n

10�12 pico p

VALUES OF TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES

q �0

�30

�37 45� �

53 60� 90�

sinq 0 1 2 3 5 2 2 4 5 3 2 1

cosq 1 3 2 4 5 2 2 3 5 1 2 0

tanq 0 3 3 3 4 1 4 3 3 �

The following conventions are used in this exam. I. The frame of reference of any problem is assumed to be inertial unless

otherwise stated. II. In all situations, positive work is defined as work done on a system.

III. The direction of current is conventional current: the direction in whichpositive charge would drift.

IV. Assume all batteries and meters are ideal unless otherwise stated.V. Assume edge effects for the electric field of a parallel plate capacitor

unless otherwise stated.

VI. For any isolated electrically charged object, the electric potential isdefined as zero at infinite distance from the charged object.

MECHANICS ELECTRICITY AND MAGNETISM

0x x xa tÃ Ã� �

x x� � 20 0Ãx t � a t

21

x

�2 20 2x x xa x xÃ Ã� � � 0

netFFa

m m� ��

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f nF Fm��

2

carÃ�

p mv��

p F tD D��

212

K mv�

cosE W F d Fd qD � � ��

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20 0

12

t tq q w a� � �

0 tw w a� �

� � cos cos 2x A t A fw� � tp

i icm

i

m xx

m�

net

I Itt

a � ��

sinr F rFt �� q

L Iw�

L ttD D�

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K Iw�

sF k x� ��

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gU mg yD D�

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si

R R� i

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p iiR R

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s iC

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energy V = electric potential v = speed k = dielectric constant r = resistivity

q = angle F = flux

MF qv B� ��

�

sinMF qv q��

B

MF I B� ��

�

sinMF I Bq��

�

B B AF � � ��

cosB B AqF ��

B

te DF

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B ve � �

�

FLUID MECHANICS AND THERMAL PHYSICS

A = areaF = force h = depth k = thermal conductivity K = kinetic energy L = thickness m = mass n = number of moles N = number of molecules P = pressure Q = energy transferred to a

system by heating T = temperature t = time U = internal energy V = volume v = speed W = work done on a system y = height�r = density

mV

r �

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�

0P P gr� � �h

bF Vgr�

1 1 2 2A v A v�

21 1

12

P gy vr� �

22 2

12

P gy vr r� � �

1r

2

kA TQt L

DD

�

BPV nRT Nk T� �

32 BK k� T

VW PD� �

U Q WD � �

MODERN PHYSICS

E = energy f = frequency K = kinetic energy � = mass p = momentum l = wavelength f = work function�

E hf�

maxK hf f� �

hp

l �

2E mc�

WAVES AND OPTICS

d = separation f = frequency or

focal lengthh = height L = distance M = magnification m = an integer n = index of

refraction s = distance � = speed l = wavelength q = angle�

vf

l �

cnÃ

�

1 1 2sin sinn nq � 2q

1 1

i os s f� � 1

i

o

hM

h� � i

o

ss

L mlD �sind mq l�

GEOMETRY AND TRIGONOMETRY

A = area C = circumference V = volume S = surface area b = base h = height � = length w = width r = radius

Rectangle A � bh

Triangle 12

A b� h

Circle 2A rp�

2C rp�

Rectangular solid V w� � h

r

Cylinder 2V rp� �

22S rp p� �� 2

Sphere 34

3V p� r

24S rp�

Right triangle 2 2c a� � b2

sin ac

q �

cos bc

q �

tan ab

q �

c a

b90�q

advanced physics course chapter 6: circular motion€¦ · advanced physics course chapter 6:...

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