Gravitational potential

Gravitational potential at a point is the work done per unit mass to bring a small mass from infinity to that point. Near a planet or star of mass M it is given by V = −GM/r, where r is the distance from the centre. It is measured in joules per kilogram (J/kg), it is zero at infinity, and it is negative everywhere else because gravity pulls masses together.

The calculator below works out V for any mass and distance, and the potential energy of a second mass placed there. It starts on Earth's surface, the same numbers as example 1.

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Gravitational Potential Calculator

Calculate gravitational potential energy

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What is gravitational potential?

Think of potential as a property of a place, not of an object. Every point around a planet has a value of V, whether or not anything is there. Put a mass m at that point and it has potential energy mV. So potential is potential energy per kilogram.

The zero is chosen at infinity, where the planet's pull has faded to nothing. Bringing a mass in from infinity, gravity does the work for you, so the mass loses energy on the way in. It ends up with less than zero, and that is why every value of V near a planet is negative.

The gravitational potential formula

The formula is for a point mass, or a uniform sphere such as a planet, measured from outside it. Measure r from the centre, not the surface: at the surface, r is the planet's radius. Potential is a scalar, so it has a size and a sign but no direction. To find the potential from two masses, add the two values of V.

How V changes with distance

Here x is the distance from Earth's centre in Earth radii and y is the potential in megajoules per kilogram (MJ/kg). Potential rises towards zero as you move out, but never reaches it. Doubling the distance halves the size of V, because V is proportional to 1/r.

Why is gravitational potential negative?

Because the zero is at infinity and gravity only attracts. To move a mass from near the Earth out to infinity, you have to do work against gravity, which raises its potential energy to zero. Something that has to gain energy to reach zero must have started below zero.

The sign also tells you which way things fall. Masses move from higher potential to lower potential when left alone, just as a ball rolls downhill. Close to the Earth, V is more negative, so that is where objects go.

Gravitational potential energy

A mass m at a point where the potential is V has gravitational potential energy:

You will also know Eₚ = mgh from GCSE. That formula gives the change in potential energy, and only near the surface, where g is roughly constant. Further out g falls off, so for satellites and planets you need the radial formula. Example 2 shows how far wrong mgh goes for a 400 km climb.

Gravitational potential difference and work done

The gravitational potential difference ΔV between two points is the difference in their potentials. The work done to move a mass m between them is:

Moving out from a planet, ΔV is positive, so you must do work. Moving in, gravity does the work and the mass gains kinetic energy instead. The path does not matter, only where you start and finish.

Equipotential surfaces

An equipotential is a surface where every point has the same potential. Around a planet they are spheres centred on it. Three facts come up in exams:

• No work is done moving along an equipotential, because ΔV = 0. A satellite in a circular orbit stays on one.
• Equipotentials are at right angles to the field lines everywhere.
• Equal steps of potential get further apart as you move out. Around the Earth, the −60, −50, −40 and −30 MJ/kg surfaces sit at 1.04, 1.25, 1.56 and 2.09 Earth radii from the centre.

Gravitational potential and field strength

Field strength g is the potential gradient, with a minus sign:

So the gradient of a V against r graph, at any point, is the size of g there. The curve above is steepest at the surface, where the gradient is 9.8 N/kg, and flattens out as g weakens. Going the other way, the area under a graph of g against r between two distances is the potential difference between them.

Here x is again the distance in Earth radii and y is g in N/kg. Because g goes as 1/r², it drops much faster than potential: at two radii g is a quarter of its surface value, while V is a half.

Worked examples

These use G = 6.674 × 10⁻¹¹ N m² kg⁻², Earth's mass 5.972 × 10²⁴ kg and Earth's mean radius 6.371 × 10⁶ m. Check any of them in the gravitational potential calculator.

Example 1: the potential at Earth's surface

The potential at the surface is −6.26 × 10⁷ J/kg, or −62.6 MJ/kg. Every kilogram on the ground would need 62.6 MJ to escape completely.

Example 2: lifting a satellite 400 km

How much energy does it take to raise a 1,000 kg satellite from the surface to a height of 400 km, ignoring the speed it needs once it gets there?

At 400 km, r = 6.371 × 10⁶ + 4.0 × 10⁵ = 6.771 × 10⁶ m, so V = −5.89 × 10⁷ J/kg. The potential difference is:

It takes 3.70 × 10⁹ J. Using mgh with g = 9.81 N/kg gives 3.92 × 10⁹ J, about 6% too much, because g is weaker at 400 km than at the surface.

Example 3: escape velocity from the potential

To escape, an object launched from the surface needs enough kinetic energy to climb all the way to V = 0. Per kilogram, that means ½v² = GM/r:

Earth's escape velocity is 11.2 km/s, whatever the mass of the object. The escape velocity calculator does this for any planet.

Example 4: Mars

Mars has a mass of 6.42 × 10²³ kg and a radius of 3.39 × 10⁶ m. Find the potential at its surface and how much energy a 50 kg lander gains, in total, when it falls to the surface from very far away.

From infinity, ΔV = −1.26 × 10⁷ J/kg, so the lander loses 50 × 1.26 × 10⁷ = 6.3 × 10⁸ J of potential energy. Without a heat shield, parachutes and engines, all of it would become kinetic energy.

Check your understanding

Common mistakes

• Measuring r from the surface. r is from the centre. For a satellite, add its height to the planet's radius.
• Dropping the minus sign. V is negative. When you find ΔV, subtract the starting value from the final one, signs and all.
• Mixing up V and Eₚ. V is per kilogram (J/kg). Multiply by the mass to get energy in joules.
• Using mgh far from the surface. It assumes g is constant, which is only true for small heights.
• Squaring r. Potential goes as 1/r. Field strength and force go as 1/r².

To test this topic under time pressure, try today's A-level Physics: Year 2 short timed paper. Gravitational fields will not be in every paper, as each one mixes topics from the year.

Read next

Physics · Fields and gravityGravitational potential calculator: V = −GM/r and UFree gravitational potential calculator: find V = −GM/r at any distance from a planet, and the potential energy U = −GMm/r of a mass placed there.