Gravity

Newton’s Law of Universal Gravitation

Newton’s insight: The force accelerating an apple downward is the same force that keeps the Moon in its orbit.

Universal Gravitation

force on due to

1 212 12 1 22

12

ˆm mF G r m m

r

Nature is nice!

1 2m m 1.00000001 1.000000011 2m m

212r

2.000000112r

The gravitational force is always attractive, and points along the line connecting the two masses:

The two forces shown are an action-reaction pair.

Ex. How strong is gravitational attraction between you and the person next to you?

G is a very small number; this means that the force of gravity is negligible unless there is a very large mass involved (such as the Earth).

Solution: 

Estimate:

1 2 12

11 72

60 ; 1

60 606.67 10 3 10

1

m m kg r m

F N

If an object is being acted upon by several different gravitational forces, the net force on it is the vector sum of the individual forces.

This is called the principle of superposition.

Gravitational Attraction of Spherical Bodies

Gravitational force between a point mass and a sphere*: the force is the same as if all the mass of the sphere were concentrated at its center.

a consequence of 1/r2

(inverse square law) *Density of sphere must be radial symmetric

Gravitational Force at the Earth’s SurfaceThe center of the Earth is one Earth radius away, so this is the distance we use:

The acceleration of gravity decreases slowly with altitude...

...until altitude becomes comparable to the radius of the Earth. Then the decrease in the acceleration of gravity is much larger:

g

In the Space Shuttle

Astronauts in Astronauts in

the space the space

shuttle float shuttle float

because:because:

a) they are so far from Earth that Earth’s gravity doesn’t act any more

b) gravity’s force pulling them inward is cancelled by the centripetal force pushing them outward

c) while gravity is trying to pull them inward, they are trying to continue on a straight-line path

d) their weight is reduced in space so the force of gravity is much weaker

In the Space Shuttle

Astronauts in Astronauts in

the space the space

shuttle float shuttle float

because:because:

a) they are so far from Earth that Earth’s gravity doesn’t act any more

b) gravity’s force pulling them inward is cancelled by the centripetal force pushing them outward

c) while gravity is trying to pull them inward, they are trying to continue on a straight-line path

d) their weight is reduced in space so the force of gravity is much weaker

Astronauts in the space shuttle float because

they are in “free fall” around Earth, just like a

satellite or the Moon. Again, it is gravity that

provides the centripetal force that keeps them

in circular motion.

Follow-upFollow-up: How weak is the value of : How weak is the value of gg at an altitude of at an altitude of 300 km300 km??

Satellite Motion: FG and acp

Consider a satellite in circular motion*:

* not all satellite orbits are circular!

Gravitational Attraction:

Necessary centripetal acceleration:

• Does not depend on mass of the satellite!• larger radius = smaller velocity smaller radius = larger velocity

Relationship between FG and acp will be important for many gravitational orbit problems

A geosynchronous satellite is one whose orbital period is equal to one day. If such a satellite is orbiting above the equator, it will be in a fixed position with respect to the ground.

These satellites are used for communications and weather forecasting.

How high are they?

RE = 6378 kmME = 5.87 x 1024 kg

Averting Disaster

a) it’s in Earth’s gravitational fielda) it’s in Earth’s gravitational field

b) the net force on it is zerob) the net force on it is zero

c) it is beyond the main pull of Earth’s c) it is beyond the main pull of Earth’s gravitygravity

d) it’s being pulled by the Sun as well as by d) it’s being pulled by the Sun as well as by EarthEarth

e) none of the abovee) none of the above

The Moon does not The Moon does not

crash into Earth crash into Earth

because:because:

The Moon does not crash into Earth because of its

high speed. If it stopped moving, it would, of course,

fall directly into Earth. With its high speed, the Moon

would fly off into space if it weren’t for gravity

providing the centripetal force.

Averting Disaster

The Moon does not The Moon does not

crash into Earth crash into Earth

because:because:

Follow-upFollow-up: What happens to a satellite orbiting Earth as it slows?: What happens to a satellite orbiting Earth as it slows?

a) it’s in Earth’s gravitational fielda) it’s in Earth’s gravitational field

b) the net force on it is zerob) the net force on it is zero

c) it is beyond the main pull of Earth’s c) it is beyond the main pull of Earth’s gravitygravity

d) it’s being pulled by the Sun as well as by d) it’s being pulled by the Sun as well as by EarthEarth

e) none of the abovee) none of the above

Two Satellites

a) a) 11//88

b) ¼b) ¼

c) ½c) ½

d) it’s the samed) it’s the same

e) 2e) 2

Two satellites A and B of the same mass Two satellites A and B of the same mass

are going around Earth in concentric are going around Earth in concentric orbits. The distance of satellite B from orbits. The distance of satellite B from Earth’s center is twice that of satellite A. Earth’s center is twice that of satellite A. What is theWhat is the ratio ratio of the centripetal force of the centripetal force acting on B compared to that acting on acting on B compared to that acting on A?A?

Using the Law of Gravitation:

we find that the ratio is .we find that the ratio is .

Two Satellites

a) a) 11//88

b) ¼b) ¼

c) ½c) ½

d) it’s the samed) it’s the same

e) 2e) 2

Two satellites A and B of the same mass Two satellites A and B of the same mass

are going around Earth in concentric are going around Earth in concentric orbits. The distance of satellite B from orbits. The distance of satellite B from Earth’s center is twice that of satellite A. Earth’s center is twice that of satellite A. What is theWhat is the ratio ratio of the centripetal force of the centripetal force acting on B compared to that acting on acting on B compared to that acting on A?A?

Note the 1/R2 factor

Gravitational Potential Energy

Gravitational potential energy, just like all other forms of energy, is a scalar. It therefore has no components; just a sign.

Gravitational potential energy of an object of mass m a distance r from the Earth’s center:

(U =0 at r -> infinity)

Very close to the Earth’s surface, the gravitational potential increases linearly with altitude:

Energy Conservation

Total mechanical energy of an object of mass m a distance r from the center of the Earth:

This confirms what we already know – as an object approaches the Earth, it moves faster and faster.

Escape Speed

Escape speed: the initial upward speed a projectile must have in order to escape from the Earth’s gravity

from total energy:

If initial velocity = ve, then velocity at large distance goes to zero. If

initial velocity is larger than ve, then there is non-zero total energy, and the kinetic energy is non-zero when the body has left the potential well

Maximum height vs. Launch speedSpeed of a projectile as it leaves the Earth, for various launch speeds

Kepler’s Laws of Orbital MotionJohannes Kepler made detailed studies of the apparent motions of the

planets over many years, and was able to formulate three empirical laws

You already know about circular motion... circular motion is just a special case of elliptical motion

1. Planets follow elliptical orbits, with the Sun at one focus of the ellipse.

Only force is central gravitational attraction - but for elliptical orbits this has both radial and tangential components

Elliptical orbits are stable under inverse-square force law.

Kepler’s Laws of Orbital Motion

2. As a planet moves in its orbit, it sweeps out an equal amount of area in an equal amount of time.

This is equivalent toconservation of angular momentum

v Δt

r

p p a a

p p a a p a

L mr v mr v

r v t r v t A A

Perigee

Apogee

Kepler’s Laws of Orbital Motion

3. The period, T, of a planet increases as its mean distance from the Sun, r, raised to the 3/2 power.

This can be shown to be a consequence of the inverse square form of the gravitational force.

Orbital Maneuvers

Which stable circular orbit has the higher speed?

How does one move from the larger orbit to the smaller orbit?

Binary systemsIf neither body is “infinite” mass, one should consider

the center of mass of the orbital motion

1 21 2

1 2

1 1 2 2

2 2

CM :

r rT T

v v

m r m r

Four equations in fourunknowns

If you weigh yourself at the equator If you weigh yourself at the equator

of Earth, would you get a bigger, of Earth, would you get a bigger,

smaller, or similar value than if you smaller, or similar value than if you

weigh yourself at one of the poles?weigh yourself at one of the poles?

a) bigger value

b) smaller value

c) same value

Guess My Weight

If you weigh yourself at the equator If you weigh yourself at the equator

of Earth, would you get a bigger, of Earth, would you get a bigger,

smaller, or similar value than if you smaller, or similar value than if you

weigh yourself at one of the poles?weigh yourself at one of the poles?

a) bigger value

b) smaller value

c) same value

The weight that a scale reads is the normal forcenormal force exerted by

the floor (or the scale). At the equator, you are in circular you are in circular

motionmotion, so there must be a net inward forcenet inward force toward Earth’s

center. This means that the normal force must be slightly less normal force must be slightly less

than than mgmg. So the scale would register something less than your

actual weight.

Guess My Weight

Earth and Moon I

a) the Earth pulls harder on the Moona) the Earth pulls harder on the Moon

b) the Moon pulls harder on the Earthb) the Moon pulls harder on the Earth

c) they pull on each other equallyc) they pull on each other equally

d) there is no force between the Earth d) there is no force between the Earth and the Moonand the Moon

e) e) it depends upon where the Moon is in it depends upon where the Moon is in its orbit at that timeits orbit at that time

Which is stronger,

Earth’s pull on the

Moon, or the

Moon’s pull on

Earth?

By Newton’s Third Law, the forces

are equal and opposite.

Earth and Moon I

a) the Earth pulls harder on the Moona) the Earth pulls harder on the Moon

b) the Moon pulls harder on the Earthb) the Moon pulls harder on the Earth

c) they pull on each other equallyc) they pull on each other equally

d) there is no force between the Earth d) there is no force between the Earth and the Moonand the Moon

e) e) it depends upon where the Moon is in it depends upon where the Moon is in its orbit at that timeits orbit at that time

Which is stronger,

Earth’s pull on the

Moon, or the

Moon’s pull on

Earth?

Principle of EquivalenceYou’re standing at rest on a scale. The display shows that it is exerting a force on you of

F = mg = 60 * 9.81 = 589 N

You’re now at rest in outer space, far from any star or planet. You fire your thruster which exerts a force of 150 N on you (m = 60 kg). At what rate do you accelerate?

F = ma so a = F/m = 150/60 = 2.5 m/s2

m and m represent two different concepts. Why can we treat them interchangeably?

General Relativity

More complete theory of gravity.

Replaces “spooky” action-at-a-distance with curvature of space, an idea that is just about as “spooky” as action-at-a-distance.

Required to make GPS work!

Black holesIf an object is sufficiently massive and sufficiently small, the escape speed will equal or exceed the speed of light – light itself will not be able to escape the surface.

This is a black hole.

The light itself has mass (in the mass/energy relationship of Einstein), or spacetime itself is curved

Gravity and lightLight will be bent by any gravitational field; this can be seen when we view a distant galaxy beyond a closer galaxy cluster. This is called gravitational lensing, and many examples have been found.