Question

If the radius of earth shrinks by 1.5 percent (mass remaining same), then the value of gravitational acceleration changes by

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

Answer 1

The correct option is C

$3\%$

Step 1. Given data:

Radius $R$ is decreased by $1.5\%$

So the new radius $R^{\text{'}}$=$R-0.015R=0.985R$………….$\left(1\right)$

Step 2. Calculations:

Let the test mass used is m and the mass of the earth is M.

Then attraction force on the test mass , $F_1$ $=\frac{GMm}{R^2}$,

The weight of the test mass is $W=mg$

As the gravitational force of attraction is the measure of the weight of the test.

Hence, we can write,

$F_1=W$

Or $\frac{GMm}{R^2}$= $mg$

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

Putting the value of $R\text{'}$ from $\left(1\right)$. we get $g\text{'}$

$g^{\text{'}}=\frac{GM}{{\left(0.985R\right)}^2}=1.0306\frac{GM}{R^2}$

So $$\begin{gathered} (g\text{'}-g)=\frac{GM}{R^2}\left(1.0306-1\right) \\ =0.0306 \end{gathered}$$

Hence percentage change=$0.0306\times 100=3.06\cong 3\%$

Thus, the correct option is option C.