Question

If you doubled the mass and tripled the radius of a planet, by what factor would g at its surface change?

Short answer

The factor by which the acceleration due to gravity changes is $\frac{2}{9}$.

Step-by-step solution

Step 1. Given Data

The increased mass of the planet is $m\text{'}=2m$.

The increased radius of the planet is $r\text{'}=3r$.

Step 2. Understanding the acceleration due to gravity

The acceleration due to gravity of the planet does get affected by the variation in the mass and radius of the planet. As the mass doubles, the acceleration due to gravity will increase by a factor of 2.

Step 3. Estimating the change in acceleration due to gravity on the planet

The relation of acceleration due to gravityis given by,

$g=G\frac{m}{r^2}$

Here, G is the gravitational constant, m is the mass and$r$ is the radius of the planet.

When the values of mass and radius are increased, the relation of acceleration due to gravity will become,

$\begin{array}{l}g\text{'}=G\frac{m\text{'}}{r{\text{'}}^2}\\ g\text{'}=G\frac{2m}{{\left(3r\right)}^2}\\ g\text{'}=G\frac{2m}{9r^2}\\ g\text{'}=\frac{2}{9}\left(g\right)\end{array}$

Thus, $\frac{2}{9}$is the factor by which the acceleration due to gravity changes.