Discussion
The ratio between the gravitational potential energies of 1 kg of mass on the surface of two planets having masses and radii in the ratio 1 : 2 and 1 : 3 respectively, is
Answer: C
The ratio is 3 : 2.
The potential energy of 1 kg on the surface is \(U = -\frac{GM}{R}\), so the ratio of magnitudes is
\[ \begin{aligned} \frac{U_{1}}{U_{2}} \\ &= \frac{M_{1}}{M_{2}}\cdot\frac{R_{2}}{R_{1}} \\ &= \frac{1}{2}\cdot\frac{3}{1} \\ &= \frac{3}{2} \end{aligned} \]
2 : 3 inverts it, and 9 : 4 and 4 : 9 use \(R^{2}\), which is right for gravitational field strength but not for potential energy.
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