# Calculating Gravitational Potential(OCR A Level Physics)

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Katie M

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Physics

## Calculating Gravitational Potential

#### Calculating Gravitational Potential

• Gravitational potential Vg can be calculated at a distance r from a point mass M using the equation:

• Where:
• Vg = gravitational potential (J kg1)
• G = Newton’s gravitational constant (N m2 kg–2)
• M = mass of the body causing the gravitational field (kg)
• r = distance from the centre of mass of M to the point in the field (m)

• This means that the gravitational potential is negative on the surface of a mass (such as a planet), and increases with distance from that mass (becomes less negative toward zero)
• Work has to be done against the gravitational pull of the planet to take a unit mass away from the planet
• The gravitational potential at a point depends on the mass of the object producing the gravitational field and the distance the point is from that mass

#### Changes in Gravitational Potential

• Two points at different distances from a mass will have different gravitational potentials
• This is because the gravitational potential increases with distance from a mass

• Therefore, there will be a gravitational potential difference between the two points
• This is represented by the symbol ΔV

• ΔV can therefore be expressed as the difference between the 'final' gravitational Vf potential and the 'initial' gravitational potential Vi

ΔV = Vf – Vi

• Therefore, the change in potential between two points a distance r1 and r2 from some mass M is given by:

• This simplifies to:

• Where:
• ΔV = change in potential (J kg–1)
• G = Newton’s gravitational constant (N m2 kg–2)
• M = mass causing the gravitational field (kg)
• r1 = initial distance from mass M (m)
• r2 = final distance from mass M (m)

#### Worked example

Calculate gravitational potential at the surface of Mars.

Radius of Mars = 3400 km

Mass of Mars = 6.4 × 1023 kg

Step 1: Write the gravitational potential equation

Step 2: Substitute known quantities

= – 1.3 × 107 J kg–1

#### Exam Tip

The equation for gravitational potential in a radial field looks very similar to the equation for gravitational field strength in a radial field, but there is a very important difference! Remember, for gravitational potential:

so

However, for gravitational field strength:

so

Additionally, remember that both Vg and g are measured from the centre of the mass M causing the field!

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