How does the weight of an object vary

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? (CBSE 2012)

Solution:

Weight of an object, $\mathrm{W}=m g=\frac{\mathrm{GM} m}{\mathrm{R}^{2}}\left(\because g=\frac{\mathrm{GM}}{\mathrm{R}^{2}}\right)$

So, weight of an object is directly proportional to the mass $(M)$ of the earth and inversely proportional to the square of the radius (R) of the earth.

If $\mathrm{D}^{\prime}=\frac{\mathrm{D}}{2}$ or $\mathrm{R}^{\prime}=\frac{\mathrm{R}}{2}$ and $\mathrm{M}^{\prime}=4 \mathrm{M} \therefore$ Weight, $\mathrm{W}^{\prime}=\frac{\mathrm{G} \times 4 \mathrm{M} m}{(\mathrm{R} / 2)^{2}}=\frac{16 \mathrm{GM} m}{\mathrm{R}^{2}}=16 \mathrm{~W}$

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