Physics 55Monday, September 26, 2005

1. Newton’s Law of Gravitation With Examples2. Connection of Newton’s Laws to Fundamental Conservation Laws

Announcements

• Answers to Assignment 3 posted.• Quiz this Friday at 1:15 pm sharp.• Homework 4 due this Friday.• Short Assignment 5 on Friday and due

Wednesday, Oct 5. Will give detailed answers on Oct. 5.

• Review session for midterm next week (to be announced).

• Midterm on Friday, Oct 7.

Questions from Previous Lecture?

• Kepler’s three laws• Newton’s three laws• Circular motion: radial force causes acceleration

Simplest Motion: Uniform Motion

Nonuniform Motion: AccelerationSpeed is not constant or direction of motion is not constant (but speed can be constant in case of circular motion).

Where are speeds large in this picture if stroboscope samples at equal times?

Where are speeds small in this picture?

a = (1/m) F

A

B

Newton’s Great Insight:Nonuniform Motion Caused by Forces

Something from one object like Sun somehow influences motion of other object like Earth. That “something” is still not understood in any fundamental sense but Newton discovered could be described by an astonishingly simple and precise mathematical rule now known as the universal law of gravitation.

The gravitational force becomes weaker with distance but has an effect no matter how far one object is from other object. Total force on object is sum of forces from all other objects so depends on relative positions of all the other objects. Mathematics can be hard, computers have helped to obtain insight.

1 2gravity 1 12

11 3 2

=

~ 6.7 10 m /(kg s )

M MF G M a

d

G

Newton’s Third Law is Built intoThe Law of Gravity Since M1 M2 = M2 M1

As strange as it may sound, you pull as hard on the Earth via gravity as the Earth pulls on you (you get the same value of F since value does not depend on order of product of masses M1 and M2).

Does not seem this way because mass of Earth is enormous and so its acceleration a=(1/m)F due to your force is not noticeable.

Henry Cavendish 1730-1810Measured G and So “Weighed the Earth”

221

d

MMGF

Newton Described GravityHe Did Not Explain Gravity!

An important point: Newton was successful because he gave up trying to explain “what is gravity” but instead tried to describe gravity mathematically and economically.

This was huge change in philosophy, say compared to people like Descartes who tried to come up with intuitive mechanisms.

Modern science has followed the path of Newton: we know how to describe and predict many phenomena with great accuracy but we don’t have an intuitive mechanistic explanation for why the description holds. This is especially frustrating with quantum mechanics, the theory of atomic particles and light.

Difference Between Mass and Weight of Object

Mass m of object is a number (units of kg) that measures resistance to motion of object caused by force acting on object. It is an intrinsic property of an object and does not change.

Example: Kicking a bowling ball in outer space can break your toe because it has a mass and so inertia and resists force from your toe.

Weight of an object is total force (a vector or arrow) that arises from gravitational pull and accelerations (for example, from elevator or amusement ride). Weight has units of newtons.

Example: If you stand on a bathroom scale in outer space far from any object, you will measure zero weight even though you certainly have a mass.

Example of Gravitational Force Causing Acceleration

Letter to Einstein From Jerry, Richmond, VA 1952

Dear Sir,

I am a high school student and have a problem. My teacher and I were talking about Satan. Of course you know that when he fell from heaven, he fell for nine days, and nine nights, at 32 feet a second and was increasing his speed every second. I was told there was a foluma [formula] to it. I know you don't have time for such little things, but if possible please send me the foluma. Thank you, Jerry

Challenge for class: given formula v= g t, with what speed did Satan hit the Earth given gravitational acceleration near surface of Earth of g ~ 9.8 m/s2? Given formula d= ½ g t2, how far did Satan fall? Note: 1 AU ~ 1.5 108 km

Solution to the Einstein LetterAnswer to the letter: Let’s assume as Jerry does that Satan falls from heaven with a constant acceleration equal to the acceleration g at the surface of the Earth and see where our thinking takes us. We don’t know if Satan was thrown from heaven with some initial speed so let’s assume the simplest case that he was pushed out of heaven and so starts falling with zero initial speed. Assuming nine full days of 24 hours each (should we use solar or sidereal days for a heavenly problem?), the final speed would be v = g t =( 9.8 m/s^2)(9 d x 24 h/d x 60 m/h x 60 s/m) ~ 8 10^6 m/s ~ 0.025c, i.e., 2.5 percent the speed of light which is really cruising (nothing orbiting within the solar system moves so quickly!). The distance traveled would be given by the formula d = (1/2) g t^2 = (1/2)(9.8 m/s^2)(9 d x 24 h/d x 3600 s/h)^2 ~ 3 10^12 m ~ 3 10^9 km ~ 20 AU or halfway to the planet Pluto, whose elliptical orbit has a semimajor axis of 40 AU.

Two punch lines: whoever originated the Satan story was not thinking big enough since it seems unlikely that heaven would lie between Neptune and Pluto. Second, a constant acceleration of one g leads to impressive speeds, about 3% the speed of light after just 9 days of travel. As you might guess, it takes a powerful rocket and a lot of fuel to maintain such an acceleration for so long.

The fact that Satan falls from a distance of 20 AU means that the assumption of constant acceleration g is wrong: for an object so far from Earth, gravity is much weaker and the acceleration GMEarth/d2 starts small and becomes bigger, slowly increasing to the value g as Satan gets close to the Earth. So the above calculation overestimates the final speed and overestimates the distance traveled. To get a more accurate estimate of Satan’s final speed and the distance Satan traveled, we need to use calculus (or a clever trick without calculus using Newton’s version of Kepler’s third law, see the extra credit problem of Assignment 4). We also need to make some further scientific assumptions, such as whether Earth is the only source of gravity acting on Satan or whether we need to include the Sun. Let’s continue to work with the assumption that Earth is the source of gravity and that hell lies within Earth (the magma?). Then I get a value for the distance fallen of 6 10 5 km ~ 1.5 x distance to Moon, much closer to Earth. The final speed of Satan at the Earth’s surface is obtained by energy conservation (decrease in potential energy must equal Satan’s kinetic energy) which I find to give a value of v ~ 11 km/s or about 34 times the speed of sound. This is fast but not beyond human technology to achieve.

Simple Worked Example of Gravity and Forces

Which ball has no weight?Which ball won’t move at all?Which ball has the largest acceleration? Which two balls will collide first?Speed of balls after 1 hour?

Note: For each ball, you need to calculate total force from two forces arising from the other two to determine a ball’s acceleration. Assume balls start at rest and pull on each other via gravity.

Gravitational Acceleration Near Earth’s Surface

2 2

2 2

or

m9.8

s

mM GMma G a

d dGM

gR

Gravitational acceleration depends on mass M of object and on distance R from center of spherical object to surface (radius).

What is acceleration on surface of Moon, for which mMoon ~ 0.012 mEarth and RMoon ~ 0.27 REarth? Answer: gMoon/gEarth = 0.012/.272 ~ 0.16. So your weight mg on the Moon is about 1/6 your weight on Earth.

Conservation Laws and Newton’s Laws

From our modern perspective, we now understand Newton’s first, second, and third laws to be statements of “conservation of momentum”.

In absence of force, momentum of object is constant so its velocity (direction and speed) is constant. This is Newton’s first law, but should have been called Galileo’s law of inertia.

Force causes momentum of object to change: either speed or direction changes. This is Newton’s second law: d/dt(mv) = sum of forces.

Since momentum is conserved, whenever two objects interact (asteroid colliding with Earth, astronaut pushing off from International Space Station), the force of first object on a second must be exactly opposite in direction and strength of the force of second object on first.

Emmy Noether (1882-1935):Conservation is Related to Symmetry

1. Translational invariance in time implies energy conservation.

2. Translational invariance in space implies momentum conservation.

3. Rotational invariance implies angular momentum conservation.

Other possible symmetries: time reversal, mirror reflection, inversion, discrete rotations, etc.

Discovered deep connection between conservation laws and symmetries of physical equations:

1. Conservation of momentum: whenever there is a collision of objects or some object separates into pieces, the total momentum has the same value. Use to predict speeds, explains how rockets work.

Points Covered at White Board

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'112211 vmvmvmvm

Las Vegas CSI Problem:Using Conservation Laws to Solve a Crime

After finishing Physics 55 and graduating from Duke, you follow your childhood dream and become a member of the Las Vegas Crime Scene Investigation (CSI) team. One day, you are called to a crime scene where you are asked to determine whether a certain rifle at the scene was used as a weapon. You can do this by determining whether the rifle is capable of firing a high-velocity bullet (mass 4 grams) in excess of 900 m/s (about 2000 mph). This is the minimum speed that could explain the substantial damage done by the bullet to a wall when the bullet fortunately missed the victim.

Not having time to get back to the lab but having some rope, a scale, and a measuring tape, you set up the following experiment: you suspend a soft block of wood (mass 0.5 kg) from a long rope so the block can swing freely back and forth. You then clear the room, make sure that the block is hanging motionless, then fire a bullet from the rifle horizontally into the wood block. (The bullet gets trapped in the soft wood.) You then observe that, from the impact of the bullet, the block swings upward along an arc to a maximum height that is 0.8 m above the initial position of the block.

I will show in class how using conservation of momentum followed by conservation of energy allows us to determine the speed of the bullet and determine whether the rifle was the weapon.

Solution to the CSI Rifle Problem