Question

How does the weight of an object vary with respect to mass and radius of the earth? In a hypothetical case, if the diameter of the earth becomes half of its present value and its mass becomes four times of its present value, then how would the weight of any object on the surface of the earth be affected?

Answer 1

1. Weight of the body is given by $W=mg$
2. Acceleration due to gravity can be rewritten as $g=\frac{GM}{R^2}$
3. Therefore, $W=\frac{GMm}{R^2}$
4. Hence, the weight of an object is directly proportional to the mass of the Earth and inversely proportional to its radius.

Hypothetical case:

Step 1: Given

Original mass$=M$

Original radius$=R$

New mass$=4M$

New radius$=\frac{R}{2}$

Step 2: Calculate the new weight

We know that,

$W=\frac{GMm}{R^2}$

Substituting new values,

$$\begin{gathered} W=\frac{G\left(4M\right)m}{{\left({\displaystyle \frac{R}{2}}\right)}^2} \\ =\frac{4GMm}{{\displaystyle \frac{R^2}{4}}} \\ =\frac{16GMm}{R^2} \\ =16W \end{gathered}$$

Hence, the new weight will be $16$ times the original weight.