Question

The masses of two planets are in the ratio 1:2. Their radii are in the ratio 1:2. The acceleration due to gravity on the planets are in the ratio:

• A. 1:2
• B. 2:1
• C. 3:5
• D. 5:3

Answer 1

The correct option is B 2:1

Let the mass of the first planet be $M_1$
Let the radius of the first planet be $R_1$
Let the mass of the second planet be $M_2$
Let the radius of the second planet be $R_2$

Acceleration due to gravity is given by:
$g=\frac{GM}{R^2}$​

For the first planet, acceleration due to gravity is:
$g_1=\frac{G{M}_1}{R_1^2}$
For the second planet, acceleration due to gravity is:
$g_2=\frac{G{M}_2}{R_2^2}$

Taking the ratio,
$\frac{g_1}{g_2}=\frac{G{M}_1/R_1^2}{G{M}_2/R_2^2}$
$\frac{g_1}{g_2}=\frac{M_1}{M_2}\times \frac{R_2^2}{R_1^2}$

It's given that $M_2=2M_1$ and $R_2=2R_1$.
On substituting these in the expression, we get:
$\frac{g_1}{g_2}=\frac{1}{2}\times \frac{2^2}{1^2}$
$\frac{g_1}{g_2}=\frac{2}{1}$