Physics 9702 · Paper 4

Graphs of g and φ

Half the marks in recent gravitational questions are graph marks: sketching the two curves, or pulling numbers off them.

The two curves

Outside the planet, g follows the inverse square law (curve sampled from g = GM/r²): ¼ g₀ at 2R, ⅑ g₀ at 3R.

g against r falls as $1/r^2$ : a quarter at 2R, a ninth at 3R. Sketch marks: starts at the surface value, always positive-curved, gradient size always decreasing (9702/41/O/N/23 Q1(b)(iii)).
φ against r rises toward zero as $-1/r$ : half as deep at 2R, a quarter at 4R (lesson 13.04's figure). Sketch marks: entirely negative, magnitude and gradient both shrinking with r, and passing through the 1/r checkpoints (9702/42/F/M/25 Q2(a)).

φ against 1/r is a straight line with gradient −GM: the modern way papers hand you a planet's mass. Read the gradient, divide by G (9702/42/F/M/24 Q1(b)(i)).
At a sphere's surface the two quantities lock together: $g = \dfrac{GM}{R^2} = \dfrac{-\phi}{R}$ . The mark scheme accepts the $\phi/R$ shortcut directly (9702/42/F/M/23 Q1(b)(iii)).
Reading a radius off a φ graph: the axis is distance from the centre. Answers that treat it as height above the surface lose the read-off mark (9702/42/F/M/23 Q1(b)(iv)).
A geostationary satellite sits at fixed r, so its φ is a horizontal line over a full day: a two-mark sketch in 2025 (9702/42/F/M/25 Q2(b)).

Remember the pair by their powers: g goes as 1/r², φ goes as −1/r. The φ curve is always the shallower one, and only φ is negative.

Graphs of g and φ

Half the marks in recent gravitational questions are graph marks: sketching the two curves, or pulling numbers off them.

Outside the planet, g follows the inverse square law (curve sampled from g = GM/r²): ¼ g₀ at 2R, ⅑ g₀ at 3R.

g against r falls as $1/r^2$ : a quarter at 2R, a ninth at 3R. Sketch marks: starts at the surface value, always positive-curved, gradient size always decreasing (9702/41/O/N/23 Q1(b)(iii)).
φ against r rises toward zero as $-1/r$ : half as deep at 2R, a quarter at 4R (lesson 13.04's figure). Sketch marks: entirely negative, magnitude and gradient both shrinking with r, and passing through the 1/r checkpoints (9702/42/F/M/25 Q2(a)).

φ against 1/r is a straight line with gradient −GM: the modern way papers hand you a planet's mass. Read the gradient, divide by G (9702/42/F/M/24 Q1(b)(i)).
At a sphere's surface the two quantities lock together: $g = \dfrac{GM}{R^2} = \dfrac{-\phi}{R}$ . The mark scheme accepts the $\phi/R$ shortcut directly (9702/42/F/M/23 Q1(b)(iii)).
Reading a radius off a φ graph: the axis is distance from the centre. Answers that treat it as height above the surface lose the read-off mark (9702/42/F/M/23 Q1(b)(iv)).
A geostationary satellite sits at fixed r, so its φ is a horizontal line over a full day: a two-mark sketch in 2025 (9702/42/F/M/25 Q2(b)).

Remember the pair by their powers: g goes as 1/r², φ goes as −1/r. The φ curve is always the shallower one, and only φ is negative.