Gravitational Potential Energy Between Two Point Masses
- The gravitational potential energy (G.P.E) at point in a gravitational field is defined as:
The work done in bringing a mass from infinity to that point
- The equation for G.P.E of two point masses m and M at a distance r is:
- The change in G.P.E is given by:
ΔG.P.E = mgΔh
- Where:
- m = mass of the object (kg)
- ɸ = gravitational potential at that point (J kg-1)
- Δh = change in height (m)
- Recall that at infinity, ɸ = 0 and therefore G.P.E = 0
- It is more useful to find the change in G.P.E e.g. a satellite lifted into space from the Earth’s surface
- The change in G.P.E from for an object of mass m at a distance r1 from the centre of mass M, to a distance of r2 further away is:
Change in gravitational potential energy between two points
- The change in potential Δɸ is the same, without the mass of the object m:
Change in gravitational potential between two points
Gravitational potential energy increases as a satellite leaves the surface of the Moon
Maths tip
- Multiplying two negative numbers equals a positive number, for example:
Worked example
A spacecraft of mass 300 kg leaves the surface of Mars to an altitude of 700 km. Calculate the change in gravitational potential energy of the spacecraft. Radius of Mars = 3400 km
Mass of Mars = 6.40 x 1023 kg
Step 1: Difference in gravitational potential energy equation
Step 2: Determine values for r1 and r2
r1 is the radius of Mars = 3400 km = 3400 × 103 m
r2 is the radius + altitude = 3400 + 700 = 4100 km = 4100 × 103 m
Step 3: Substitute in values
ΔG.P.E = 643.076 × 106 = 640 MJ (2 s.f.)
Exam Tip
Make sure to not confuse the ΔG.P.E equation with
ΔG.P.E = mgΔh
The above equation is only relevant for an object lifted in a uniform gravitational field (close to the Earth’s surface). The new equation for G.P.E will not include g, because this varies for different planets and is no longer a constant (decreases by 1/r2) outside the surface of a planet.
