Question

If the Earth shrinks half in radius and its mass remains the same, then by what percentage will the weight of an object on Earth change?

• A. Decreases by $50\%$
• B. Increases by $50\%$
• C. Decreases by $25\%$
• D. Increases by $300\%$

Answer 1

The correct option is D Increases by $300\%$

Before it shrinks, Weight of object on Earth $\left(W\right)=\frac{G{M}_{e}m}{R_e^2}$ $\cdots \left(1\right)$
When the Earth shrinks i.e. $R_e^{\prime }=\frac{1}{2}{R}_e$ , Weight of object on Earth $W^{\prime }=\frac{G{M}_{e}m}{{\left(\frac{1}{2}{R}_e\right)}^2}$
$W^{\prime }=4\frac{G{M}_{e}m}{R_e^2}$ $\cdots \left(2\right)$
W' = 4W from equation (1)
Percentage change in weight of an object
$=\frac{4W-W}{W}\times 100=\frac{3W}{W}\times 100=300$
So, weight of the object increases by $300\%$.