Understanding of definition of gravitational potential, including zero…

The idea

Gravitational potential, V, at a point is the work done per unit mass in bringing a small test mass from infinity to that point. Because the gravitational force is attractive, the work done by the field is positive when moving from infinity, so the potential is negative near a mass and increases towards zero at infinity. The defining equation is V = -GM/r for a point mass or uniform sphere, where G is the gravitational constant, M is the mass producing the field, and r is the distance from the centre. The zero value is chosen at infinity, where the field strength is zero. For example, at the surface of the Earth, using M = 5.97 × 10²⁴ kg and r = 6.37 × 10⁶ m gives V ≈ -6.25 × 10⁷ J kg⁻¹. Potential is a scalar, so values add algebraically.

What earns the marks

• Defines gravitational potential as work done per unit mass in moving a small test mass from infinity to the point.
• States that the zero of gravitational potential is at infinity.
• Explains that gravitational potential is negative near a mass because the gravitational force is attractive and the field does positive work as the test mass moves from infinity.
• Uses V = -GM/r correctly for a point mass or uniform sphere, with r measured from the centre.
• Recognises that gravitational potential is a scalar quantity and that potentials from multiple masses add algebraically.
• Calculates potential at a point using given values of G, M and r, including correct substitution and units J kg⁻¹.

Common mistakes and how to put them right

• Treating gravitational potential as a vector and attempting to resolve it into components.

  Put it right: Treat it as a scalar and add values algebraically.

• Omitting the negative sign in V = -GM/r.

  Put it right: Include the negative sign because the potential is negative near a mass when zero is at infinity.

• Confusing gravitational potential with gravitational field strength.

  Put it right: Remember that potential is work done per unit mass (J kg⁻¹) while field strength is force per unit mass (N kg⁻¹).

Exam tips

• When defining gravitational potential, always include the phrase 'work done per unit mass' and specify the path from infinity.
• In calculations, write the equation V = -GM/r, substitute values with units, and give the answer in J kg⁻¹ with an appropriate number of significant figures.
• If a question asks about the zero of potential, state clearly that it is at infinity and explain that this choice makes the potential negative near a mass.