Use gravitational potential energy for point masses
Connect potential to the energy of a test mass, use Ep = −GMm/r and interpret energy changes between orbital radii.
A Level; Cambridge International Physics 9702, 2025–2027 syllabus, Topic 13. These original explanations and calculations are study aids; full timed assessment and human subject-teacher review remain outstanding.
Before you begin
- Use Newton's laws, circular motion, vectors, scientific notation and SI units.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
How do you obtain a test mass's gravitational potential energy from potential?
What you will learn
- Relate gravitational potential to potential energy per unit mass
- Use Ep = −GMm/r for two point masses
- Calculate energy changes relative to infinity
Potential energy is mass multiplied by potential
A test mass m at potential φ has gravitational potential energy Ep = mφ. Substituting φ = −GM/r gives Ep = −GMm/r for two point masses with zero potential energy at infinite separation.
The energy is negative for a bound pair because work must be supplied to separate the masses to infinity. The equation uses centre-to-centre distance and masses in kilograms.
Compare positions by taking an energy difference
When a mass moves from r₁ to r₂, its energy change is ΔEp = Ep₂ − Ep₁. Moving outward makes the value less negative, so ΔEp is positive and energy must be supplied against gravity.
Near Earth's surface, mgΔh is a useful approximation for small height changes. The point-mass formula describes larger changes in distance and uses the zero at infinity; do not mix the two reference conventions without calculating a difference.
Pause and explain
What is Ep for two positive point masses at a finite separation using zero at infinity?
Worked example
Find the gravitational potential energy of a 2.0 kg mass at radius 2.0 × 10⁷ m from a point mass 6.0 × 10²⁴ kg.
Show the worked solution
- Use Ep = −GMm/r.
- Substitute −6.67 × 10⁻¹¹ × 6.0 × 10²⁴ × 2.0 / 2.0 × 10⁷.
- Give joules and preserve the sign.
Answer Ep ≈ −4.0 × 10⁷ J.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Calculate Ep for masses 3.0 kg and 4.0 kg separated by 0.50 m.
Give me a hint
Use −Gm₁m₂/r, distinguish this energy in joules from potential in J kg⁻¹.
Compare my reasoning
- Ep = −Gm₁m₂/r.
- Ep = −(6.67 × 10⁻¹¹)(3.0)(4.0)/0.50.
- Evaluate in joules.
Ep ≈ −1.6 × 10⁻⁹ J.
Look for these in your work
- Both masses included
- Separation not squared
- Negative energy and unit stated
A satellite moves from r₁ = 2.0 × 10⁷ m to r₂ = 4.0 × 10⁷ m around a mass 6.0 × 10²⁴ kg. Find the change in Ep for a 2.0 kg satellite.
Give me a hint
Calculate each negative energy, then subtract initial energy from final energy.
Compare my reasoning
- Ep₁ = −GMm/r₁ ≈ −4.00 × 10⁷ J.
- Ep₂ = −GMm/r₂ ≈ −2.00 × 10⁷ J.
- ΔEp = Ep₂ − Ep₁ ≈ +2.00 × 10⁷ J.
The potential energy increases by about 2.0 × 10⁷ J as the satellite moves outward.
Look for these in your work
- Both endpoint energies calculated
- Final minus initial used
- Positive outward change interpreted
A student says negative gravitational potential energy means the satellite has negative mass. Correct the interpretation.
Give me a hint
Separate the sign of a chosen energy reference from the sign of the physical mass.
Compare my reasoning
- The masses are positive.
- Zero potential energy is defined at infinite separation.
- At finite separation the bound pair has lower energy, so Ep is negative; separating it requires positive work.
Negative Ep describes energy below the zero-at-infinity reference, not negative mass.
Look for these in your work
- Mass sign distinguished
- Reference explained
- Energy meaning corrected
Common mistakes
- Confusing potential with potential energy
- Dropping the negative sign on a bound pair's energy
- Using mgΔh for a change comparable with the planet's radius
What can you explain now?
How do you obtain a test mass's gravitational potential energy from potential?
Compare with the explanation
Multiply potential by the mass
Ep = mφ, with joules as the resulting unit.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
Why does moving a satellite outward from a planet increase its gravitational potential energy?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.