Question

If the earth shrinks half in radius, its mass remaining 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%

Acceleration due to gravity: The acceleration achieved by any object due to the gravitational force of attraction by the earth is called acceleration due to gravity by the earth.

As each planet has a different mass and radius so the acceleration due to gravity will be different for a different planet.
Acceleration due to gravity of the earth having mass $\left(M\right)$ and radius $\left(R\right)$ on earth surface is given by: $g=\frac{GM}{R^2}$.

Here $G$ is the universal gravitational constant.

Weight:

Before it shrinks, weight of object on Earth is given by: $W=\frac{\mathrm{GMm}}{{\mathrm{R}}^2}-------\left(1\right)$

When the Earth shrinks i.e. $R\text{'}=\frac{R}{2}$ then weight of object on Earth is given by: $W^{\text{'}}=\frac{GMm}{{\left(\frac{R}{2}\right)}^2}$

$$\begin{gathered} W^{\text{'}}=4\frac{GMm}{R^2}-------\left(2\right) \\ W\text{'}=4W(U\sin gequation(1\left)\right) \end{gathered}$$

Percentage change in weight of an object becomes

$$\begin{gathered} =\frac{4\mathrm{W}-\mathrm{W}}{\mathrm{W}}\times 100 \\ =\frac{3\mathrm{W}}{\mathrm{W}}\times 100 \\ =300\% \end{gathered}$$

So, weight of the object increases by $300\%.$

Hence, option (D) is correct.