(c) force–distance graph for a point or spherical mass; work done is area…

The idea

For a point mass or a spherical mass treated as a point at its centre, the gravitational force on a test mass varies with distance r as F = GMm/r², an inverse-square relationship. A force–distance graph plots this force on the vertical axis against separation r on the horizontal axis, producing a curve that falls steeply as r increases. The work done by the gravitational field in moving the test mass between two separations equals the area under the force–distance graph between those limits, with the sign indicating whether the field does work or work is done against it. Because the curve is not linear, the area must be found by integration, by counting squares, or by using the potential difference: work done per unit mass equals the change in potential.

For example: The area under F = GMm/r² from r = R to r = 2R gives the energy released as the mass falls inward.

What earns the marks

• Sketchs or interprets a force–distance graph showing F = GMm/r² decreasing with increasing r.
• States that work done is represented by the area under the force–distance graph between two separations.
• Recognises that the area cannot be found by a simple triangle or rectangle rule because the curve is non-linear.
• Links the area under the graph to the change in gravitational potential energy, or per unit mass to the change in potential.
• Applies the result to a point mass or to a spherical mass outside its surface, using r from the centre.

Common mistakes and how to put them right

• Treating the force–distance graph as a straight line and using ½ × base × height.

  Put it right: F ∝ 1/r² gives a curve, so integration or square-counting is needed.

• Confusing the area under a force–distance graph with the gradient.

  Put it right: Gradient gives force per unit distance, while area gives work done.

• Using the surface radius instead of the centre-to-centre separation for a spherical mass.

  Put it right: r is measured from the centre when outside the sphere.

Exam tips

• Label both axes with quantity and unit, and mark the two separation limits between which the area is required.
• If a numerical area is needed, state the method used, such as integration of GMm/r² or counting squares.
• Use the link work done = mΔV to check an area obtained graphically.