Planet N has a gravitational potential -1/V at its surface - AQA - A-Level Physics - Question 16 - 2018 - Paper 2

To determine the gravitational potential at the surface of planet M, we can employ the following relationships:

1. Relation between mass and density: The density of a planet is defined as:

   $\rho = \frac{m}{V}$ where $m$ is the mass and $V$ is the volume. For a spherical planet, the volume is given by:

   $V = \frac{4}{3} \pi r^3$

   Consequently, the mass $m$ is:

   $m = \rho \cdot V = \rho \cdot \frac{4}{3} \pi r^3$

2. Planet M's parameters: Since planet M has double the density and double the radius of Planet N:

   $\rho_M = 2\rho_N$ $r_M = 2r_N$

3. Find mass of Planet M: Substituting these values into the mass equation gives:

   $m_M = (2\rho_N) \cdot \frac{4}{3} \pi (2r_N)^3 = 16 \cdot \frac{4}{3} \pi r_N^3 \rho_N = 16 m_N$ Thus, the mass of planet M is 16 times the mass of planet N.

4. Gravitational potential relation: The gravitational potential $V$ at the surface of a sphere is given by:

   $V = -\frac{G m}{r}$ where $G$ is the gravitational constant. For planet M:

   $V_M = -\frac{G (16 m_N)}{(2r_N)} = -\frac{16}{2} \cdot \left(-\frac{G m_N}{r_N}\right) = -8 \left(-\frac{1}{V}\right) = -8 V$

Therefore, the gravitational potential at the surface of planet M is (-8 V), leading us to the answer.