# PHY C8: Gravitational Potential

We’re covering:

• Gravitational Potential
• Gravitational Potential Energy
• Change in Gravitational Potential Energy
• Total energy of a satellite
• Escape velocity

Let’s go!

What is Gravitational Potential?
Work done per unit mass in bringing a small mass from infinity to a certain point.

It is denoted by φ (phi).

φ = W/m

Since W = Fr
W/m = (GMm/r2) x r/m
W/m = GM/r*

*However, by definition (see below), we take φ as always being a NEGATIVE value.

So,
φ = – GM/r

Why is it always negative?
There are a few ways to explain this:

 By Definition We define infinity to be our starting point. At infinity, F = 0, g = 0, φ = 0. To bring a mass from infinity to a point IN a gravitational field, we do NEGATIVE work (instead of lifting the mass against the gravitational force, we let it ‘fall’ naturally). By Calculation Work done is the integral of Fg. ∫ Fg dr = ∫ GMm/r2 dr W = – GMm/r φ = W/m φ = – GM/r By applying the Principle of Conservation of Energy As the test mass moves from infinity to a point within the gravitational field, it LOSES the potential to do work (i.e. it can do less work). In this sense it LOSES potential energy, & gains kinetic energy.

What is Gravitational Potential Energy?
Work done in bringing a small mass from infinity to a certain point.

Ep = mφ
Ep = – GMm/r

Remember: this is the energy needed to bring a mass m from infinity to a point within a gravitational field.

What is the work done to move an object between 2 points within a gravitational field?
Work done = Change in Gravitational Potential Energy

ΔEp = EpA – EpB

If the change in distance is small enough such that change gravitational field strength is negligible, we can actually say:
ΔEp = mgh

This is where that expression comes from!

What is the total energy of an orbital satellite?
Kinetic Energy + Gravitational Potential Energy.

ΣE = Ek + Eg
ΣE = ½ mv2 + (- GMm/r)

Since mv2/r = GMm/r2 (Fc = Fg),
½ mv2 = GMm/2r

Plugging this in:
ΣE = GMm/2r + (- GMm/r)
ΣE = – GMm/2r

What is escape velocity?
The minimum velocity required to escape a gravitational field.
The “edge” of a gravitational field is where the field strength g approaches 0.

Thus, an object which has escaped does not accelerate towards M.

When an object is launched from the surface at the escape velocity,
it will decelerate due to Fg until it reaches a velocity of 0 ms-1 at the edge of the gravitational field.

Since the object is launched from the surface,

• Initial Eg = -GMm/R
• Final Eg = -GMm/ = 0
• Gain in Eg = GMm/R

Loss of Ek = Gain in Eg
½ mve2 = GMm/r

ve = √(2GM/R)

OR

ve = √(2gR)