A planet has radius $R$ and density $\rho$ - AQA - A-Level Physics - Question 12 - 2021 - Paper 2

To find the gravitational field strength at the surface of the planet with new radius and density, we can use the formula for gravitational field strength:

$g' = \frac{GM}{R^2}$

1. Calculate the new mass:

   • The volume of the planet is given by:
     $V = \frac{4}{3}\pi R^3$
   • For the new planet with radius $2R$:
     $V' = \frac{4}{3}\pi (2R)^3 = \frac{4}{3}\pi (8R^3) = 8 \cdot \frac{4}{3}\pi R^3$
   • The new mass $M'$ with density $2\rho$:
     $M' = V' \cdot 2\rho = 8 \cdot \frac{4}{3}\pi R^3 imes 2\rho = \frac{64}{3}\pi R^3 \cdot \rho$

2. Calculate the new gravitational field strength:

   • Substituting for $M'$ and $R$ into the gravitational field strength formula:
     $g' = \frac{G \cdot \frac{64}{3}\pi R^3 \cdot \rho}{(2R)^2}$
   • Simplifying:
     $g' = \frac{64}{3} \cdot \frac{G \cdot \pi \cdot \rho R^3}{4R^2} = \frac{16}{3} \cdot \frac{G \cdot \pi \cdot \rho R^3}{R^2} = \frac{16}{3} g$

We find that the new gravitational field strength is:
$g' = 4g$
Therefore, the correct answer is B) $4g$.