Question

If the radius of earth shrinks by $1.5\%$ (mass remaining same), then the value of gravitational accelaration changes by

• A. $2\%$
• B. $-2\%$
• C. $3\%$
• D. $-3\%$

Answer 1

The correct option is C $3\%$

Given:
$-\frac{\Delta R}{R}=1.5\%$ ($-ve$ sign for shrink)

The acceleration due to gravity on the surface of a planet is given by

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

Taking $\log$ and differentiating on both sides, we get:

$\Rightarrow \frac{\Delta g}{g}=-2\frac{\Delta R}{R}$
$[G$ and $M$ are constant $]$

$\Rightarrow \frac{\Delta g}{g}=2\times 1.5\%=3\%$

Hence, option $\left(c\right)$ is constant.

Why the question: To make students understand the quantities on which the acceleration due to gravity depends. Also Try: Keep radius constant and find out the variation with the mass of density of the planet.