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Gravitation

Newton's Law of Universal Gravitation

Fg = GM1M2 / R2

Since Fg = m * a, we have

a = Fg / m = GM/ Re2

Where Me is earth's mass, and Re is radius of earth in meters.

G is gravitational constant, G = 6.67 x 10-11 N * m2 / kg2

a = 9.8 m/s2 = g

Note that the value of g is not constant but varies inversely with the earth radius squared Re2

Gravitational Energy

Potential energy of an object on earth with mass M0

PE = -GM0Me / Re (from Fg = GM1M2 / R2 and E = F * Δd)

Since a = Fg / m = GM/ Re2 = g

We have

PE = - M0 * a * Re = - M0 * g * Re

From above, when we lift an object with mass m from the ground by a height of h, the work done will be:

ΔPE = mgh

Escape Velocity

The escape velocity is the minimum initial velocity required for an object to leave a planet, for example, earth.

(1/2)M0Vesc2 = GM0Me / Re

Vesc2 = 2 G Me / Re

Note escape velocity depend on the mass of the earth but not on the mass of the object!

Note: this section is not required in SAT Physics

Orbit Velocity

The orbit velocity can be determined by assuming that we have an approximately circular orbit. In this case, the centripetal force equals the gravitational force:

GM0Me / Re= M0 Vorbit2 / Re

Vorbit2 = G Me / Re

Note orbit velocity depend on the mass of the earth but not on the mass of the object.

Note: this section is not required in SAT Physics

Kepler's Laws

http://hyperphysics.phy-astr.gsu.edu/hbase/kepler.html