Question

A planet reduces its radius by $1\%$ keeping its mass unchanged. How does its acceleration due to gravity change?

Answer 1

Step 1 :Given data and Formula used

Given a planet reduces its radius by $1\%$ keeping its mass unchanged.

As we know acceleration due to gravity(g) =$GM/R^2$

Where,

G=universal gravitation constant,

M= mass of planet,

R= radius

Step 2 : Finding the change in the value of g

we have, $g=GM/R^2$

$g\text{'}=GM/R{\text{'}}^2$

where, $R\text{'}=(R-\frac{R}{100})=0.99R$

We have to put value of R' in terms of R

$$\begin{gathered} g\text{'}=\frac{GM}{{(0.99R)}^2} \\ g\text{'}=1.02\frac{GM}{R^2} \end{gathered}$$

Change in acceleration due to gravity is

$$\begin{gathered} g\text{'}-g=(1.02-1)\frac{GM}{R^2} \\ =0.02\frac{GM}{R^2} \end{gathered}$$

3.To calculate percentage:

$$\begin{gathered} 0.02\frac{GM}{R^2}\times \left(100\right) \\ =2\% \end{gathered}$$

Hence acceleration due to gravity increases by $2\%$ as radius decreases.

Hence, the correct answer is 2%.