9.2 – Gravitational Potential and Escape Velocity

Gravitational Potential (V) in a Uniform Field

As you lift a mass above the ground you do work.

The mass gains PE=mgh

PE mass∝ PE m∝ PE where it is ∝ PE gh∝

The ‘gh’ bit of this relationship is called Gravitational Potential (V)

It tells you the potential energy of a point in space and depends only on the height (h) above the ground and the gravitational field strength (g).

Gravitational Potential =J/kg

Assuming g=10Nkg-1

Gravitational Potential:VA=30 Jkg-1

VC= 140 Jkg-1

Work done moving 5kg from A to C is: 550J

Lines of Equipotential & Field Lineseg. near the surface of the earth:

Lines of Equipotential:These just join up points with the same V.

They are basically the same as contour lines

Field Lines :These show which way a mass will move when placed in the field

Where the lines of equipotential are closest the field is strongest

Lines of Equipotential & Potential Gradient

Because the field strength is largest where the lines of equipotential are closest this gives us another way to measure the field strength:

Derivation: As you move a mass upwards from A to B:

Work Done = change in potential ⨯ mass (ΔV.m)Work Done = force ⨯ distance (mg.Δh)

mgh = ΔVm

𝑔=∆𝑉∆h

ΔhVB=80 Jkg-1

VA=30 Jkg-1

m

The Field Strength (g) is the same as the Potential Gradient (

Gravitational Potential (V) due to large spheres

The Gravitational Potential at a point P is defined as the work done per unit mass taking a small test mass from ZERO potential to point P.

The only place with ZERO potential has to be INFINITY (∞).

The reason it has to be negative is that you would do work pushing the mass away from the planet to ∞.

Gravitational Potential (V) Equation

𝐹=𝐺𝑀𝑚𝑟2

The area under a force-distance graph is the same as the Work Done so it can be used to calculate Potential.

The equation is:

𝑉=−𝐺𝑀𝑟

Negative because the force is to move from ∞ to r

Lines of Equipotential and Potential Wells

Lines of equipotential around a sphere.

Visualising them as a surface helps us to understand why gravity makes things move. This picture is called a potential well.

Relationship between field lines and potential

This picture has field lines and lines of equipotential.

Remember they are always perpendicular to each other (at right angles).

The lines of equipotential are closest where the field is strongest.

The field is strongest where the lines of equipotential are most dense.

Adding Potentials

Adding potentials for two masses involves a simple calculation.

Escape Speed – you need to learn this derivation:

This is the speed to escape the gravitational attraction of a planet or mass.

On Earth is something is thrown upwards for it to escape it most get from the Earth’s surface () to infinity where PE = 0.

Loss of KE Gain in PE

Loss of KE PE at ∞ PE at Surface12𝑚𝑣2=𝐺

𝑀𝑚𝑟 𝐸 𝑣𝑒𝑠𝑐𝑎𝑝𝑒=√ 2𝐺𝑀𝑟 𝐸

On Earth this is 11 kms-1

This is why the Moon has lost it’s atmosphere. The escape velocity for a molecule was similar to the speed of the molecules in its early atmosphere.

Escape Speed simulator:

http://bit.ly/EscapeSpeed OR CLICK ON THE PICTURE