Chapter 10 - Gravity and Motion

Newton’s First Law of Motion

• A body continues in a state of rest or uniform motion in a straight line unless made to change that state by forces acting on it.

• The natural behavior of objects is to continue to move however they have been moving (inertia).

• Any time a body changes how it is moving, there is always some force that caused that change.

Orbital Motion and Gravity

• A force is any kind of push or pull exerted by one object on another.

• Besides contact, friction, electric, magnetic, elastic, pressure, etc. forces, Newton said that objects also exert a gravitational force on each other.

• The force of gravity causes all bodies to attract all other bodies.

• Gravity, coupled with laws of motion, enabled Newton to explain exactly how orbits work.

The Moon and the Earth

• We just learned that the Earth will exert a gravitational forces on the Moon pulling the Moon towards the Earth.

• So, what holds the Moon up?• Why doesn’t it fall down like

if you drop a rock?

Earth

Moon

GravitationalForce

The Moon and the Earth (2)

• If the Moon was just sitting up there, it would fall straight down onto the Earth.

• But the Moon is moving, “sideways” at a pretty high speed.

• The Moon does fall down, but it is moving sideways at the same time.

• Just like if I throw a baseball, it moves across the room while falling downwards.

Earth

Moon

GravitationalForce

Moon’s velocity

The Moon and the Earth (3)

• Without gravity, the Moon would move in a straight line, flying away from the Earth.

• The orbit is a balance between the natural straight-line motion and the attractive pull towards the Earth.

• The Moon is always falling towards the Earth but it is also always shooting away from the Earth.

Earth

Moon

GravitationalForce

Moon’s velocity

Path followed by the Moon

The Sun and Planets

• Orbits of planets around the Sun work just like the Moon’s orbit around the Earth.

• If the gravitational force and orbital speed are exactly balanced, a planet will orbit in a perfect circle.

• If the planet’s speed is a little faster or slower, a non-circular orbit results.

• If the planet’s speed is much too fast or slow it may escape the Sun altogether or fall into the Sun.

Unfinished Details

• More details about: exactly how gravity works, the shapes of orbits, and escaping from the gravity of a planet will all be discussed later in the chapter.

The Earth, a Pencil, and Gravity

• The Earth exerts a force of gravity on a pencil causing it to fall (accelerate) to the floor, but clearly the pencil does not exert an equal force on the Earth! Right?

• The pencil moves but the Earth just sits there.• The forces are equal! That does not mean the

accelerations are equal. a = F/m• The Earth has a mass 1027 times more than the

pencil, so for the same force it has 1027 times less acceleration, immeasurably small.

The Law of Gravity

• The three laws of motion would’ve been enough to make Newton famous, but that’s just a small fraction of his accomplishments.

• He also derived a law to explain gravity.• Called the “Law of Universal Gravitation”

because he applied the law to both objects on Earth and to astronomical objects like the Moon, Sun, and planets.

Law of Gravitation

• Every mass exerts a force of attraction on every other mass. Further, the strength of the force is directly proportional to the product of the masses divided by the square of their separation.

• The force of gravitational attraction is given by:

F = G M m / r2 where

F is the gravitational force (in newtons)

M, m are the masses of the attracting bodies (in kilograms)

r is the distance between the (centers) of the bodies

G is a proportionality constant that depends on units

Gravitational Constant G

• In the usual metric units,

G = 6.67 x 10-11 N m2 / kg2

• Fear not, you will not have to memorize this number or these units, the value and units will be supplied every time it is needed - which will be quite a bit.

I know, all these dry equations and laws are boring. So here’s a change of pace, we will resume the usual

boring lecture momentarily.

Intermission

Law of Gravity Example

Example: Calculate the force of gravity exerted by the Earth on a 7-kg bowling ball.

Solution: F = GMm/r2

G = 6.67 x 10-11 Nm2/kg2

M = MEarth = 6 x 1024 kgm = Mball = 7 kgr = distance between centers = radius of Earth = 6,378,000 m

F = (6.67 x 10-11)(6 x 1024)(7)/(6,378,000)2 N

= 68.9 N (kg and m units all cancelled out)

Second Law Example

Example: If the bowling ball from the previous example is dropped, how fast will it accelerate?

Solution: a = F/m

F = 68.9 N

m = 7 kg

a = (68.9)/(7) = 9.8 m/s2

The Earth also feels the force of 68.9 N but accelerates much less (essentially zero acceleration) because of its far larger mass.

Acceleration of Falling Objects

• To calculate the force of gravity acting on the bowling ball, we had to multiply by 7 kg (the m in F = GMm/r2).

• Then to get the acceleration due to this gravitational force we divided by 7 kg (the m in a = F/m).

• So the 7 kilograms didn’t matter. We would get the same acceleration for any object with any mass at the surface of the Earth, 9.8 m/s2.

Galileo’s Experiment• Aristotle had claimed that

heavier objects fell faster than lighter ones. Twice the weight would fall twice as fast he said.

• Galileo did experiments that easily proved this was not true (although that he did the experiments/demonstration at the Leaning Tower of Pisa is believed to be a myth).

Universal Laws

• How did Newton decide that this was the right law of gravity? (F = GMm/r2)

• Because only this particular equation explained both objects moving on Earth and the motions of planets (elliptical orbits, equal areas in equal times).

• Newton derived the correct law of gravity from Kepler’s laws.

• In so doing, he then discovered that Kepler’s laws had some slight inaccuracies.

Newton’s Model

• Newton had his own answer to the geocentric/ heliocentric debate.

• He said neither the Earth nor the Sun is at the center of the universe.

• Newton believed the universe to be infinite and center-less with everything moving around.

• The infinite universe remains a viable picture of the universe today.

Perturbations

• The other part of Kepler’s first law was also modified, planets do not move along perfect ellipses.

• Planets “perturb” (deflect or nudge) each other with their gravity, causing tiny wiggles in the paths the planets follow (only measurable using telescopes).

• Newton’s laws allow exact predictions of these perturbations.

Jupiter

Mars

Sun

Elliptical Orbit

Perturbed orbit due to Jupiter

Measuring Mass Using Orbital Motion

• We’ve learned that a lot of things are connected.– The masses of bodies and their separation determines

how much gravitational force they exert on each other.

– The force on a body and its mass determines what acceleration it will experience.

– Acceleration is the rate of change of velocity, so knowing how big an orbit a planet is following and how long it takes to orbit would allow you to calculate what acceleration it is undergoing.

Gravitational Force

• The gravitational force between objects is directly proportional to the mass of the particles and inversely proportional to the square of the distance between them.

• F= mxM / d2

• F=Force, m=mass of object, M=mass of earth/mass of second object, d=distance apart

• The quantity of the universal constant of gravitation, G must be inserted

• G = 6.67x10-11 Nm2/kg2

Gravitational Force

• Which gives us,– Force = G x m1 x M2 / d2

– Gravitational force is measured in newtons, like all other forces

• Examples– The gravitational force of attraction between the Earth

and the sun is 1.6x1023. What would be the force if Earth were twice as big?

• F=Gm1M2 / d2

• = 2(1.6x1023N) • = 3.2x1023N

Gravitational Force Example

• Ben, whose mass is 85kg, sits 2.0m apart from Nick, whose mass is 100kg. What is the force of attraction between Ben and Nick? Why don’t Ben and Nick drift toward one another?– Force = G x m1 x m2 / d2

– = (6.67x10-11 Nm2/kg2)(85kg)(100kg) / 2m2

– = 1.41x10-7 N

Gravitational Force Example

• When Sammy was 10 years old, she had a mass of 22kg. By the time she was 16 she then weighed 45kg. How much larer is the gravitational force on earth at age 16 compared to age 10?– Force = G x m1 x m2 / d2

– = (6.67x10-11 Nm2/kg2)(22)– = 1.46x10-9

– Force = G x m1 x m2 / d2

– = (6.67x10-11 Nm2/kg2)(45)– = 3.00x10-9

Gravitational Acceleration

• You can use the law of universal gravitation to find the gravitational acceleration, g, of any body if you know that body’s mass and radius.

• g = GM2 / d2

• d here is equal to the radius of the celestial body

Gravitational Acceleration

• Andy Ford is standing in the lunch line at 6.38x106m from the center of the earth. The earth’s mass is 5.98x1024kg. What is his acceleration due to gravity?

• g = GM2 / d2

• = (6.67x10-11 Nm2/kg2)(5.98x1024kg) / 6.38x106m

• = 9.8m/s2

Gravitational Acceleration

• What is the acceleration due to gravity on the sun if the radius of the sun is

• g = GM2 / d2

Escape Velocity

• Throw an object upwards: it goes up, stops, falls back down.

• Throw the object up with a faster speed and it will go higher before falling back.

• Throw an object fast enough (called the escape velocity) and Earth’s gravity is not enough to stop it and bring it back. It escapes into space.

• Guess what? You can use Newton’s laws to derive a formula for escape velocity! You want to know what the formula is? Okay…

Escape Velocity Formula

• Escape velocity vesc2 = 2 G m / r where

G = gravitational constant

M = mass of planet

R = radius of planet

• Example:• Calculate the escape velocity from Earth

given that the mass of Earth is 6 x 1024 kilograms and its radius 6 x 106 meters.

Example: Earth’s Escape Velocity• Solution:

• Escape velocity vesc2 = 2 G M / R

G = 7 x 10-11 m3/(kg s2)

m = 6 x 1024 kg

r = 6 x 106 m

vesc2 = 2 (7e-11) (6e24) / (6e6) m2/s2

= 11,832 m/s

= 11,832 m/s (1 km/1000 m) = 11.8 km/s

End of Lecture Review

• What should you have learned?– How gravity and inertia cause orbits

– Newton’s law of gravity

– The Sun moves and planets perturb each other

– That orbital properties can be used to find masses

– What “surface gravity” and “escape velocity” are

– How to calculate gravitational force, gravitational acceleration, and escape velocity