GRAVITATIONKeplers Laws1. Planets move in elliptical orbits with the sun at one focus.2. As a planet moves in its orbit, a line drawn from the sun to the planet sweeps

out equal areas in equal time intervals.3. If T is the period and a is the length of a the semi-major axis, then T2/a3 is a

constant

Newtons Universal Law of Gravitation- any massive objects in the universeexert an attractive force on each other called the gravitational force.

Fg= 221

rmGm G = 6.67 x 10-11 Nm2/kg2

Where m1 is the mass of the first object, m2 is the mass of the second object andr is the distance between their centers of masses. And Fg is the gravitational forcebetween them. G is the universal gravitation constant.

The force is negative because it is an attractive force.

Using this equation, you can determine the acceleration of gravity on the surfaceof a planet by setting m2g= 2

21

rmGm and solving for g.

m2will cancel out, m1 is the mass of the planet and r is the distance from thecenter of mass of the planet.

If there are more than 2 objects, you can find the net force on one object byfinding its gravitational attraction to each of the masses surrounding it and thenperforming a vector summation of those forces. Here it could be helpful to writethe forces as unit vectors ( i and j).

In the equation, the masses are directly proportional to the force. This meansthat if one mass is doubled, the force between the two objects is doubled. Animportant part of this equation comes from the fact that r2 appears in thedenominator. This is known as an inversed-square law.

Circular OrbitsIf we can assume a circular orbit for a satellite in orbit around a planet (or aplanet around a star) we know that the gravitational force between them iscausing the circular motion. In other words, centripetal force = gravitationalforce.

Fcentripetal = Fgravitation

m1v2/r= 221

rmGm

where m1 is the mass of the satellite in orbit and m2 is the mass of the planet, andr is the distance between the objects centers.

Using these equations, you can obviously solve for any of the variables included,but you can also solve for the satellites period. Period, T, is the time it takes forthe satellite to complete one full revolution.

T = vr2

Gravitational Fields

Spherical Shell When you are outside of the shell, it can be treated as if all themass is at the center, a distance r away.

When you are inside a spherical shell, you experience no net gravitational force.g(r) = 0 Fg = 0 for r< R

g(r) =2r

Gm Fg = 221

rmGm for rR

Gravitational Force vs. Distance for A SphericalShell r

Fgrav

itatio

n

R

Solid Sphere When you are outside of the sphere, it can be treated as if all themass is at the center, a distance r away.

When you are inside a solid sphere, the mass in the shell above you exerts no netgravitational force (like above). However, the mass in the sphere below youdoes exert a force. So when you are a distance r away from the center, youexperience a gravitational force due to the mass and radius of the massbeneath you acting as its own planet.

g(r) =3R

Gm r Fg = 321

RmGm r for r< R

g(r) =2r

Gm Fg = 221

rmGm for rR

The equations for r< R come from a more general equation

Fg= 2inm1

rGm side In this space, solve for Fg = 3

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RmGm r

using Newtons Universal Law of gravitation. Hint: use density to put minside interms of known variables such as the mass of the planet and the radius of theplanet.

Gravitational Force vs. Distance for A SolidSphere r

Fgrav

itatio

n

R

Gravitational Potential Energy

U = - - 2 21r mGm dr

U = - rmGm 21

This means that the gravitational potential energy is negative at the surface ofthe planet and becomes closer and closer to zero until you are an infinitedistance away, in which case the gravitational potential energy is zero.

You will want to use this equation instead of U= mgh when you are dealing with aproblem where the projectile is far from the surface of the Earth (when g is nolonger 9.8 m/s2).

Gravitational potential energy is also helpful when solving problems involving theconservation of mechanical energy, such as when a projectile is launched awayfrom the surface of the Earth or when a projectile in space comes towards theEarth.

Total mechanical energy E= Uo + Ko

Uo + Ko = Uf + KfInstead of using mgh here, make sure you use the new form.

-ormGm 21 + m vo2 = -

frmGm 21 + m vf2

(make sure that you use the distance from their center of masses for r).

Escape Speed- the minimum speed required to escape from the Earths gravity.

If Ko > Uo then it will escape (unbounded)If Ko < Uo then it wont escape (bounded)

So if E = Ko + Uo 0, than the objects can escape!

-ormGm 21 + m vo2 0

thenormGm 21 = m vo2

so vescape = RGmearth2

Newtons Universal Law of Gravitation- any massive objects in the universe exert an attractive force on each other called the gravitational force.