Calculating gravitational potential: $V = -GM/r$

Gravitational Fields & Orbits — AQA A-Level Physics

V = -\frac{GM}{r}

$V$: gravitational potentialThe work doneEnergy transferred when a force moves an object. In electrical circuits, W = QV (chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C). times potential difference). per unit mass in bringing a small test mass from infinity to that point. Always negative. Measured in J kg⁻¹. (J kg⁻¹)
$G$: Newton's gravitational constant (6.67 × 10⁻¹¹ N m² kg⁻²)
$M$: mass producing the gravitational fieldA region of space in which a mass experiences a gravitational force. (kg)
$r$: distance from the centre of the mass (m)

As r decreases (moving towards the planet), V becomes more negative.
As r increases (moving away), V becomes less negative, tending to zero at infinity.
The gravitational potentialThe work doneEnergy transferred when a force moves an object. In electrical circuits, W = QV (chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C). times potential difference). per unit mass in bringing a small test mass from infinity to that point. Always negative. Measured in J kg⁻¹. difference between two points is $\Delta V = V_{f} - V_{i}$.

Worked Example

A planet has diameter 7600 km and mass 3.5 × 10²³ kg. Calculate the gravitational potential at 400 km above the surface.

Show Solution

Write the equation

$$V = -\frac{GM}{r}$$

Calculate r from the centre

Radius of planet = 7600 / 2 = 3800 km

$$r = 3800 + 400 = 4200 \text{ km} = 4.2 \times 10^6 \text{ m}$$

Substitute values

$$V = -\frac{(6.67 \times 10^{-11}) \times (3.5 \times 10^{23})}{4.2 \times 10^6}$$

Evaluate

$$V = -5.6 \times 10^6 \text{ J kg}^{-1}$$

Answer

$V = -5.6 \times 10^6$ J kg⁻¹

Examiner Tips and Tricks

Remember to calculate r from the centre of the planet, not just the distance above the surface.
Add the planet's radius to the altitude.