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Universal Gravitation Lecturer Professor

Stephen T. Thornton

Reading Quiz

The International Space Station is at an altitude

of 200 km above the surface of the Earth. What

is the net force on an astronaut at rest inside

the Space Station?

- Equal to her weight on Earth.
- A little less than her weight on Earth.
- Less than half her weight on Earth.
- Zero (she is weightless).
- Somewhat larger than her weight on Earth.

Answer B

- The astronaut is falling around the Earth. The

gravitational force is keeping her from going in

a straight line. The acceleration of gravity is

a little less than g at 200 km above the Earth.

200 km

6380 km

Last Time

- Non-uniform circular motion
- Drag
- Terminal velocity
- Fundamental forces

Today

- History of gravitation
- Newtons law of universal gravitation
- Keplers laws

History of Gravitation

- Greeks used the geocentric frame in which the

Earth was at the center. - Ptolemy, 2nd A.D., prepared a detailed

formulation of heavenly body motion. - epicycles
- retrograde motion

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? Nicolaus Copernicus (1473-1543 ) introduced

heliocentric frame with the Sun at the center

of the solar system.? Catholic church thought

this was heresy.? Danish astronomer Tycho Brahe

(1546- 1601) made huge number of observations.

Johannes Kepler continued his work and did

analysis.

Galileo (1564-1642) is said to have invented the

telescope. He made many observations.Issac

Newton (1642-1727) was one of the smartest

persons to ever live.He invented calculus so he

could solve the problem of how the moon rotates

around Earth.Newton looked at Keplers results

and figured everything out in a manner of days!

Newtons Law of Universal Gravitation

If the force of gravity is being exerted on

objects on Earth, what is the origin of that

force?

Newtons realization was that the force must come

from the Earth. He further realized that this

force must be what keeps the Moon in its orbit.

conic sections

Newton showed that planetary motion and other

similar motion had to be of conic sections.

Newtons Law of Universal Gravitation

- The force of gravity between any two point

objects is attractive and of magnitude

G is the universal gravitational constant

Gravitational Force Between Point Masses

Important points about gravity

- Gravitational force is a vector and always

attractive. - Gravity is difficult to measure, except for large

bodies. - Gravity has an exact 1/r2 dependence.
- For several masses, just add forces. Called

superposition.

Gravitational Force Between a Point Mass and

a Sphere (uniform mass density)

Use symmetry. For this case, force acts at

center.

Gravitational Force Between

the Earth and the Moon

Conceptual Quiz

A) the Earth pulls harder on the Moon B) the

Moon pulls harder on the Earth C) they pull on

each other equally D) there is no force between

the Earth and the Moon E) it depends upon

where the Moon is in its orbit at that time

- Which is stronger, Earths pull on the Moon, or

the Moons pull on Earth?

Conceptual Quiz

A) the Earth pulls harder on the Moon B) the

Moon pulls harder on the Earth C) they pull on

each other equally D) there is no force between

the Earth and the Moon E) it depends upon where

the Moon is in its orbit at that time

- Which is stronger, Earths pull on the Moon, or

the Moons pull on Earth?

By Newtons 3rd Law, the forces are equal and

opposite.

Conceptual Quiz

A) one quarter B) one half C) the same D)

two times E) four times

- If the distance to the Moon were doubled, then

the force of attraction between Earth and the

Moon would be

Conceptual Quiz

A) one quarter B) one half C) the same D)

two times E) four times

- If the distance to the Moon were doubled, then

the force of attraction between Earth and the

Moon would be

The gravitational force depends inversely on the

distance squared. So if you increase the

distance by a factor of 2, the force will

decrease by a factor of 4.

Follow-up What distance would increase the

force by a factor of 2?

Force of gravity on a mass m on the surface of

the Earth is mg.Lets use Newtons universal law.

Keplers three laws follow naturally from

Newtons Law of Universal Gravitation.First law

occurs because of 1/r2. Orbits must be ellipses.

We will derive Keplers laws later after we

study angular momentum.

Keplers 1st and 2nd laws

http//physics.bu.edu/duffy/semester1/semester

1.html

2nd law Radius vector sweeps out equal areas in

same time interval.

Keplers 3rd law follows directly from the form

of the gravitational force law.

Keplers Third Law and Some Near Misses

Free Fall

- Look at http//galileoandeinstein.physics.virginia

.edu/more_stuff/Applets/newt/newtmtn.html

Gravitational Attraction. Two objects attract

each other gravitationally with a force of

when they are 0.25 m apart. Their

total mass is 4.00 kg. Find their individual

masses.

Suns Mass Determination. Determine the mass

of the Sun using the known value for the period

of the Earth and its distance from the Sun.

Hint The force on the Earth due to the Sun is

related to the centripetal acceleration of the

Earth. Compare your answer to that given in

your textbook.

Conceptual QuizThe gravitational constant

G is A) equal to g at the surface of Earth.

B) different on the Moon than on Earth.

C) obtained by measuring the speed of

falling objects having different masses.

D) all of the above. E) none of the above

Answer E

- None of them determine G.

Newtons Law of Universal Gravitation

Using calculus, we can show Particle outside a

thin spherical shell gravitational force is the

same as if all mass were at center of

shell. Particle inside a thin spherical shell

gravitational force is zero. See next slide. Can

model a sphere as a series of thin shells

outside any spherically symmetric mass,

gravitational force acts as though all mass is at

center of sphere.

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