(b) gravitational potential V g r GM =- at a distance r from a point mass…

The idea

Gravitational potential at a point is the work done per unit mass in bringing a small test mass from infinity to that point, giving V = −GM/r for a point mass M at distance r. The negative sign shows that potential is zero at infinity and decreases as you approach the mass, so a test mass falls spontaneously towards M. For a spherical mass of uniform density, the same expression applies outside the sphere, with r measured from the centre. A change in potential ΔV = V_final − V_initial equals the work done per unit mass moving between two points without change in kinetic energy.

For example: Moving from r = 2R to r = R near a planet changes V from −GM/2R to −GM/R, a change of −GM/2R, meaning potential decreases and gravitational potential energy is released.

What earns the marks

• States V = −GM/r for a point mass M at distance r, with r measured from the centre of the mass.
• Explains that gravitational potential is defined as work done per unit mass in bringing a test mass from infinity to the point.
• Interprets the negative sign: V is zero at infinity and becomes more negative closer to M.
• Calculates a change in potential as ΔV = V_final − V_initial, e.g. −GM/R − (−GM/2R) = −GM/2R.
• Applies the expression outside a spherical mass, treating it as a point mass at its centre.

Common mistakes and how to put them right

• Omitting the negative sign and writing V = GM/r.

  Put it right: Potential is negative because work is done by the field as the test mass moves in from infinity.

• Measuring r from the surface rather than the centre of the spherical mass.

  Put it right: r is the distance from the centre when the point lies outside the sphere.

• Treating a change in potential as a simple subtraction of magnitudes.

  Put it right: Subtract signed values, V_final − V_initial, so the sign of ΔV is meaningful.

Exam tips

• Write the defining equation with its negative sign before substituting numbers, so the sign is not lost in rearrangement.
• When asked for a change in potential, show both potential values with their signs and subtract them explicitly.
• Check that r is larger than the radius of any spherical mass before using the point-mass expression.