- The equation for gravitational potential V is defined by the mass M and distance r:
- V = gravitational potential (J kg-1)
- G = Newton’s gravitational constant
- M = mass of the body producing the gravitational field (kg)
- r = distance from the centre of the mass to the point mass (m)
- The gravitational potential always is negative near an isolated mass, such as a planet, because:
- The potential when r is at infinity (∞) is defined as 0
- Work must be done to move a mass away from a planet (V becomes less negative)
- It is also a scalar quantity, unlike the gravitational field strength which is a vector quantity
- Gravitational forces are always attractive, this means as r decreases, positive work is done by the mass when moving from infinity to that point
- When a mass is closer to a planet, its gravitational potential becomes smaller (more negative)
- As a mass moves away from a planet, its gravitational potential becomes larger (less negative) until it reaches 0 at infinity
- This means when the distance (r) becomes very large, the gravitational force tends rapidly towards 0 at a point further away from a planet
Gravitational potential increases and decreases depending on whether the object is travelling towards or against the field lines from infinity
A planet has a diameter of 7600 km and a mass of 3.5 × 1023 kg.
A rock of mass 528 kg accelerates towards the planet from infinity.
At a distance of 400 km above the planet’s surface, calculate the gravitational potential of the rock.
Step 1: Write the gravitational potential equation
Step 2: Determine the value of r
- r is the distance from the centre of the planet
Radius of the planet = planet diameter ÷ 2 = 7600 ÷ 2 = 3800 km
r = 3800 + 400 = 4200 km = 4.2 × 106 m
Step 3: Substitute in values
Remember to keep the negative sign in your solution for the gravitational potential at a point. However, if you’re asked for the ‘change in’ gravitational potential, no negative sign needs to be included since you are finding a difference in values and just the magnitude is normally required.
Remember to also calculate r as the distance from the centre of the planet, and not just the distance above the planet’s surface