IB Physics HLTopic 4 — Force FieldsPaper 1 & 2Ep = −Gm1m2/r~17 min read
The GPE Equation
Last page we said what gravitational potential energy means in a radial field, and how to dig it out of a graph. Now we write it down. And the good news is that you have already done the hard part: potential was the work done per kilogram to come in from infinity. Potential energy is that same bill, multiplied by the number of kilograms you actually brought. One line of algebra, and the equation is yours.
📘 What you need to know
Potential energy of two point masses a distance r apart: Ep = −Gm1m2/r
It is simply potential × mass: Ep = mVg
Work done moving a mass between two points: ΔW = mΔVg
Change in GPE between two radii: ΔEp = Gm1m2(1/r1 − 1/r2)
And the same without the moving mass: ΔVg = Gm1(1/r1 − 1/r2)
The change in GPE is the work done against the field
Ep is always negative (zero at infinity), a scalar, measured in joules
Both masses appear. There is no g in this equation, because g is no longer a single number
Moving away from a planet: Epincreases (work done on the mass). Moving in: Epdecreases
From potential to potential energy
Potential Vg told you the energy cost per kilogram. If you carry m kilograms instead of one, you pay m times as much.
Everything you learned about potential transfers straight across. Multiply by m, and J kg−1 becomes J.
Gravitational potential energy of two point massesEp = − G m1m2 / rm1 = mass producing the field (kg) • m2 = mass moving in it (kg) • r = separation of centres (m)
Compare it with Newton’s law of gravitation, F = Gm1m2/r². Same two masses, same G — but one power of r instead of two, and a minus sign out front. That is not a coincidence: multiplying a force by a distance gives you an energy, and it cancels one r from the bottom. Force goes as 1/r²; energy goes as 1/r. If you ever forget which is which, ask yourself which one has units of newtons.
Work done = change in GPE
Here is the sentence that unlocks most exam questions on this topic: the work done against the field is exactly the change in gravitational potential energy. Nothing is lost, nothing is hidden.
If you know the potential at two points, the work done on a mass m moving between them is:
Work done from a potential differenceΔW = m ΔVgwhere ΔVg = Vfinal − Vinitial
And if instead you are given two distances, subtract the two potential energies. Watch what happens to the minus signs:
Change in GPE between two radiiΔEp = Ep2 − Ep1 = (−Gm1m2/r2) − (−Gm1m2/r1)ΔEp = Gm1m2( 1/r1 − 1/r2)
Notice the minus sign has vanished, and the 1/r1 now comes first. That is not a typo — it is the two negatives cancelling. Drop the same trick on potential alone and you get the version without the moving mass:
Change in potentialΔVg = Gm1( 1/r1 − 1/r2)
Reading it off the curve
Draw Ep against r and the whole story is one picture. The curve sits below the axis, deepest near the planet, rising towards zero. Pick two radii, and the vertical gap between the curve’s two heights is the energy you must supply to climb from one to the other.
Zero energy is reached only at infinity. B is further out and higher up the curve, even though both values are negative. Climbing from A to B costs energy; that cost is ΔEp.
Which way did it move?
Work is done on the mass when it moves against the field lines — that is, away from the planet. Gravity does the work when it falls back in.
The satellite never escapes the negative region. It just moves to a shallower part of the well.
Notice how ΔEp = Gm1m2(1/r1 − 1/r2) polices this for you. Move out, so r2 > r1, which makes 1/r1 the bigger term, so the bracket is positive and ΔEp is positive. Energy went in. Move in, and the bracket flips negative all by itself. The equation is doing your thinking — as long as you put r1 (the start) first.
Quantity
Equation
Unit
Depends on the moving mass?
Force
F = Gm1m2/r²
N
Yes
Field strength
g = GM/r²
N kg−1
No
Potential
Vg = −GM/r
J kg−1
No
Potential energy
Ep = −Gm1m2/r
J
Yes
Potential Vg = −GM/r
multiply by the mass m
Energy Ep = −GMm/r
subtract two positions
Work done ΔEp
🛰️ Attacking a GPE question
Total energy, or a change? “The GPE of the satellite” wants Ep = −Gm1m2/r. “Work done” or “energy needed” wants ΔEp.
Both radii from the centre. Add the planet’s radius to any altitude, every single time.
Label r1 as the start and r2 as the finish. Then use ΔEp = Gm1m2(1/r1 − 1/r2) exactly as written.
Given potentials instead of radii? Use ΔW = mΔVg, with ΔVg = Vfinal − Vinitial.
Sanity check the sign. Moved outwards? ΔEp must be positive. Moved inwards? Negative.
WE 1
A satellite of mass 1200 kg orbits at a distance of 8.0 × 10⁶ m from the centre of the Earth (ME = 5.97 × 10²⁴ kg). Calculate its gravitational potential energy.
Step 1 — total energy, so use the full equationEp = −Gm1m2 / rStep 2 — substitute (one r, not r²)Ep = −(6.67 × 10⁻¹¹)(5.97 × 10²⁴)(1200) / (8.0 × 10⁶)Ep = −4.78 × 10¹⁷ / 8.0 × 10⁶Ep = −6.0 × 10¹⁰ JNegative, as it must be. This is the energy you would have to give the satellite to carry it off to infinity and leave it at rest — 60 gigajoules of it.
WE 2
A satellite of mass 800 kg is raised from the Earth’s surface to an orbit 1.20 × 10⁷ m from the Earth’s centre. Taking RE = 6.37 × 10⁶ m and ME = 5.97 × 10²⁴ kg, calculate the work done.
Step 1 — the work done is the change in GPEΔEp = Gm1m2 (1/r1 − 1/r2)Step 2 — r1 is the start: the surfacer1 = 6.37 × 10⁶ m r2 = 1.20 × 10⁷ mStep 3 — do the bracket first1/r1 − 1/r2 = 1.570 × 10⁻⁷ − 0.833 × 10⁻⁷ = 7.37 × 10⁻⁸Step 4 — multiply throughΔEp = (6.67 × 10⁻¹¹)(5.97 × 10²⁴)(800) × (7.37 × 10⁻⁸)ΔEp = 2.3 × 10¹⁰ JPositive, because the satellite climbed. Do the bracket before you multiply by the huge numbers — it is where the precision lives, and where rounding early ruins the answer.
WE 3
A probe of mass 250 kg moves from a point where the gravitational potential is −3.6 × 10⁷ J kg⁻¹ to a point where it is −1.2 × 10⁷ J kg⁻¹. Calculate the work done on the probe, and state whether it moved towards or away from the planet.
Step 1 — find the change in potentialΔV = Vfinal − Vinitial = (−1.2 × 10⁷) − (−3.6 × 10⁷)ΔV = +2.4 × 10⁷ J kg⁻¹Step 2 — multiply by the massΔW = mΔV = 250 × (2.4 × 10⁷)ΔW = 6.0 × 10⁹ JStep 3 — which way?
The potential rose towards zero, so the probe is
further from the planetBoth potentials are negative, yet ΔV came out positive — because −1.2 is greater than −3.6. Say it in words: it got closer to zero, so it moved away.
💡 Top tips
One r, not r². If you square it you have written the force equation by accident.
Total GPE is negative. A change in GPE can be either sign — that is how you know which way it moved.
In ΔEp = Gm1m2(1/r1 − 1/r2) there is no minus sign in front. It cancelled.
Put r1 = where it started. Then the sign of the answer looks after itself.
Evaluate the bracket first, to full precision, then multiply.
There is no g anywhere in this equation. If you have written one, you have slipped back to mgΔh.
“Work done” and “change in GPE” are the same number. Answer one and you have answered both.
⚠ Common mistakes
Confusing ΔEp = Gm1m2(1/r1 − 1/r2) with Ep = mgΔh — they look similar and describe quite different situations
Squaring the r, giving the force instead of the energy
Keeping the minus sign in front of the ΔEp bracket, so the answer comes out backwards
Writing the bracket as (1/r2 − 1/r1) and reporting a negative energy for a climb
Using an altitude as r instead of adding the planet’s radius
Reporting a positive number for the total GPE of an orbiting satellite
Rounding the bracket to one significant figure — the two terms are nearly equal, so the answer collapses
Forgetting that Vg needs only the planet’s mass, while Ep needs both masses
Quick recap: Potential energy is potential times mass: Ep = mVg = −Gm1m2/r — a scalar, in joules, always negative, falling off as 1/r. The work done moving a mass between two points is the change in GPE: ΔEp = Gm1m2(1/r1 − 1/r2), or ΔW = mΔVg if you are handed potentials. Climb outwards and ΔEp is positive; fall inwards and it is negative.
Look back at the two graphs you have now drawn: g against r, and Vg against r. They are not independent pictures of the field — one is hiding inside the other. Ask how steeply the potential curve falls at a point, and you will find you have calculated the field strength there. That link is called the potential gradient, and it is the next page.
Losing marks on the signs?
Book a free meeting and we’ll drill GPE, potential differences and work-done questions until the minus signs behave.
