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?

Formula for the weight of an object: $\boxed{W=mg}$

$W\rightarrow$ Weight

$m\rightarrow$ mass of the object

$g\rightarrow$ acceleration due to gravity

We know that $g=G\frac{M_e}{R^2}$

Here, $G\rightarrow$ gravitational constant

$M_e\rightarrow$ mass of the earth

$R\rightarrow$ radius of the earth

So, weight of the object on earth, $W=m\times G\frac{M_e}{R^2}$

Or $W=G\frac{M_e m}{R^2}$

So, the weight of an object on earth is directly proportional to the masses of the earth and inversely proportional to the square of the radius of the earth.

If the diameter of the earth becomes half, then its radius will also become half$(\frac{R}{2})$ and the mass becomes four times, then the mass of the earth will be $4M_e$.

So, weight of the earth $W'=G\frac{4M_em}{(\frac{R}{2})^2}$

$=16G\frac{Mem}{R^2}$

$=16W$

Thus the mass of the object will become 16 times if the diameter of the earth becomes half of its present value and its mass becomes four times of its present value.

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Updated on: 10-Oct-2022

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