How does the weight of an object vary with respect to the 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 its present value,then how would the weight of any object on the surface of the Earth be affected?

(N/A) The weight of an object is directly proportional to the mass of the Earth $(M)$ and inversely proportional to the square of the radius of the Earth $(R)$.
Weight of a body $\propto \frac{M}{R^2}$.
The original weight is $W_0 = mg = G \frac{Mm}{R^2}$.
In the hypothetical case,the new mass $M' = 4M$ and the new radius $R' = \frac{R}{2}$ (since the diameter is halved,the radius is also halved).
The new weight $W_n$ is given by:
$W_n = G \frac{M'm}{(R')^2} = G \frac{(4M)m}{(\frac{R}{2})^2} = G \frac{4Mm}{\frac{R^2}{4}} = 16 \times (G \frac{Mm}{R^2}) = 16 W_0$.
Therefore,the weight of the object will become $16$ times its original value.