#### 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?

The weight of an object is nothing but the gravitational force by earth on that body.

By universal law of gravitation, if two particles of mass m1 and m2 are kept at separation then the gravitational force between them will be given as:

$F= \frac{Gm_{1}m_{2}}{r^{2}}$
The weight is generally given as mg, therefore,

$mg= \frac{GM_{e}m}{R_{e}^{2}}\\ \Rightarrow g=\frac{GM_{e}}{R_{e}^{2}}$

If we change the mass of earth and its radius (Diameter), the value of g changes, hence, the weight of the object changes.

$\Rightarrow g=\frac{GM_{e}}{R_{e}^{2}}$

The new value of g will be:

$\Rightarrow g'=\frac{G4M_{e}}{\left (\frac{R_{e}}{2} \right )^{2}}\\ \Rightarrow g'=\frac{16GM_{e}}{R_{e}^{2}}\\ \Rightarrow g'=16g$

Therefore, weight of the body will be sixteen times.