Gravitation

Gravitational Force and Field

Newton proposed that a force of attraction exists between any two masses.This force law applies to point masses not extended massesHowever the interaction between two spherical masses is the same as if the masses were concentrated at the centres of the spheres.

Newton´s Law of Universal Gravitation

Newton proposed that

“every particle of matter in the universe attracts every other particle with a force which is directly proportional to the product of their masses, and inversely proportional to the square of their distance apart”

This can be written as F = G m1m2

r2

Where G is Newton´s constant of Universal GravitationIt has a value of 6.67 x 10-11 Nm2kg-2

Gravitational Field Strength

A mass M creates a gravitational field in space around it.If a mass m is placed at some point in space around the mass M it will experience the existance of the field in the form of a gravitational force

We define the graviational field strength as the ratio of the force the mass m would experience to the mass, mThat is the graviational field strength at a point, is the force exerted per unit mass on a particle of small mass placed at that point

The force experienced by a mass m placed a distance r from a mass M isF = G Mm

r2

And so the gravitational field strength of the mass M is given by dividing both sides by mg = G M

r2

Field Strength at the Surface of a Planet

If we replace the particle M with a sphere of mass M and radius R then relying on the fact that the sphere behaves as a point mass situated at its centre the field strength at the surface of the sphere will be given byg = G M

R2

If the sphere is the Earth then we haveg = G Me Re

2

But the field strength is equal to the acceleration that is produced on the mass, hence we have that the acceleration of free fall at the surface of the Earth, gg = G Me Re

2

Gravitational Potential Energy

We have been usingTo calculate the change in

gravitational energy for objects close to the surface of the earth.

This doesn’t work if we are talking about objects that are not close to the surface.

hmgU g

Gravitational Potential Energy

Now we have to develop a new formula

2

1

r

r

gravg FdrWU 2r

mGmF E

122

2

1r

mGm

r

mGmdr

r

mGmU EE

r

r

Eg

r

mGmU Eg

Gravitational Potential Energy Curve

Escape Speed

An object can escape earth’s orbit when:

So: gUK

R

mGmmv E22

1

R

GMv

2

Motion of SatellitesAn object shot horizontally will follow a parabolic pathIf it is moving fast enough then the curvature of the earth matters.If you get the perfect speed it will just fall in a circle around the earth.

Motion of SatellitesIn order for a satellite to orbit the earth in a circular orbit, the force of gravity must be equal to the force necessary for circular motion at that orbital radius.

Set them equal and solve for speed

2R

mGmF Eg

R

mvFc

2

R

Gmv E

Notice that the speed does not depend on the mass of the satellite, but the speed does depend on the radius of the orbit.

Period of Rotation for a Circular Orbit

Period is the time taken for the satellite to make one complete orbit.

T

Rv

2

R

Gm

T

R

2EGm

RT

2/32

R

Gmv E

Kepler’s Laws of Planetary Motion

Each planet moves in an elliptical orbit, with the sun at one focus of the ellipse.A line from the sun to a given planet sweeps out equal areas in equal times.The period of the planets are proportional to the 3/2 power of the major axis lengths of their orbits.

1st Law: Elliptical Orbits

Actual orbits of planets in our solar system mostly circularWe approximate them as circular to save time.

2nd Law: Equal Areas in Equal Times

Derives from constant angular momentum.Also keeps total energy constant. Greatest Kinetic

Energy when Gravitational Energy is the most negative.

3rd Law: Period of Elliptical Orbit

We derived this relationship for circular orbits. You could do the same for elliptical orbits.

a is the length of the semi-major axisGm

aT

2/32