A planet of radius $R$ and mass $M$ has a gravitational field strength of $g$ at its surface - AQA - A-Level Physics - Question 9 - 2022 - Paper 2

Equating the two expressions for gravitational field strength: $4 \left( \frac{GM}{R^2} \right) = \frac{GM'}{R'^2}$ Dividing both sides by $G$ and rearranging gives: $4M = \frac{M'R^2}{R'^2}$ This can be rewritten to find a relationship between the mass and radius of the new planet: $M' = 4M \frac{R'^2}{R^2}$

We can evaluate the given options:

1. Row A: Radius $2R$, Mass $2M$:

   • $M' = 4M \frac{(2R)^2}{R^2} = 4M \cdot 4 = 16M$, (incorrect)

2. Row B: Radius $\frac{R}{\sqrt{2}}$, Mass $\frac{M}{2}$:

   • $M' = 4M \frac{(\frac{R}{\sqrt{2}})^2}{R^2} = 4M \cdot \frac{1}{2} = 2M$, (incorrect)

3. Row C: Radius $R$, Mass $\frac{M}{\sqrt{2}}$:

   • $M' = 4M \frac{R^2}{R^2} = 4M$, (incorrect)

4. Row D: Radius $\frac{R}{\sqrt{2}}$, Mass $2M$:

   • $M' = 4M \frac{(\frac{R}{\sqrt{2}})^2}{R^2} = 4M \cdot \frac{1}{2} = 2M$, (correct)

From the evaluation, Row D provides a condition in which the gravitational field strength is $4g$.