Explain why the gravitational field strength at the surface of a planet is not the same as the gravitational field strength at a point high above its surface, and outline how the concept of gravitational potential energy accounts for the work done in moving a mass between these two points.
Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).
A spacecraft of mass m is launched from the surface of a planet of mass M and radius R to an orbital altitude h above the surface.
Model answer (4 marks)
Gravitational field strength g decreases with distance from the planet’s centre because g∝1/r². At the surface r=R, so g=GM/R². At altitude h the distance is r=R+h, giving g=GM/(R+h)², which is smaller than at the surface.
Gravitational potential energy E_p is the work required to bring a mass from infinity to that point. It is negative and its magnitude decreases as r increases: E_p=−GMm/r. Moving the spacecraft from the surface to altitude h changes its potential energy by ΔE_p=E_p(h)−E_p(surface)=−GMm/(R+h)−(−GMm/R). This ΔE_p is positive, so work must be supplied to lift the spacecraft.
Thus the weaker field at altitude means less attractive force, and the increase in potential energy accounts for the work done against gravity.
Gravitational potential energy E_p is the work required to bring a mass from infinity to that point. It is negative and its magnitude decreases as r increases: E_p=−GMm/r. Moving the spacecraft from the surface to altitude h changes its potential energy by ΔE_p=E_p(h)−E_p(surface)=−GMm/(R+h)−(−GMm/R). This ΔE_p is positive, so work must be supplied to lift the spacecraft.
Thus the weaker field at altitude means less attractive force, and the increase in potential energy accounts for the work done against gravity.
Examiner tips
- State g∝1/r² and compare r=R vs r=R+h
- Explain E_p as work from infinity and its sign
- Show ΔE_p calculation and note it is positive
Common mistakes
- Confusing g at altitude with g at surface, ignoring r² dependence
- Treating gravitational potential energy as positive everywhere
- Omitting the negative sign in E_p or the change in sign when calculating ΔE_p
Mark scheme (4 marks)
- Gravitational field strength g is inversely proportional to the square of the distance from the centre of the planet (g ∝ 1/r²), so g decreases as distance from the centre increases.
- At the surface the distance from the centre is R, whilst at altitude h it is (R + h), which is greater, so the field strength at altitude h is smaller than at the surface.
- Gravitational potential energy (E_p) is defined as the work done per unit mass / the work done against the gravitational field in moving a mass from infinity to that point; it is negative and becomes less negative (increases) as distance from the planet increases.
- The work done in moving the spacecraft from the surface to altitude h equals the change in gravitational potential energy: W = ΔE_p = −GMm/(R+h) − (−GMm/R), which is positive, meaning energy must be supplied to the spacecraft.
Key terms in this question
gravitational field strength · gravitational potential energy · work done
Related
- All IB DP Physics Higher Level (2023 syllabus) revision notes →
- How to answer a "Explain" question →
- Decode the mark scheme abbreviations →
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