Question

If the mass of the earth decreases by 25% and its radius increases by 50% then the acceleration due to gravity at its surface decreases by nearly:

• A. 89%
• B. 67%
• C. 33%
• D. 11%

Answer 1

The correct option is B

67%

Step 1. Given data:

Initial mass = $M$

Initial radius = $R$

New mass $M_1$= $\left(\raisebox{1ex}{$75$}\!\left/ \!\raisebox{-1ex}{$100$}\right.\right)M$

New radius $R_1$ = $150\%$ of $R$ = $\left(\raisebox{1ex}{$150$}\!\left/ \!\raisebox{-1ex}{$100$}\right.\right)R$

Step 2. Formula used:

We know $g=\frac{GM}{R^2}$…………$\left(1\right)$

Step 3. Calculations:

Let $g\text{'}$ be the new acceleration due to gravity. Then putting new values in $\left(1\right)$, we get

$g\text{'}$ = $\frac{G\left({\displaystyle \frac{75}{100}}M\right)}{{\left({\displaystyle \frac{150}{100}}R\right)}^2}=\frac{GM}{R^2}\left(\frac{75}{100}\times \frac{10000}{150\times 150}\right)=g\left(\frac{5}{15}\right)=0.33g$

Hence the acceleration due to gravity decreases by = $g-g\text{'}$

$$\begin{gathered} =g-0.33g \\ =0.67g \end{gathered}$$

In terms of percentage, we can write

$0.67\times 100=67\%$

Therefore, the acceleration due to gravity decreases by nearly $67\%$.

So, the correct option is option (B).