Question

Suppose (God forbid) due to some reason, the earth expands to make its volume eight-fold. What do you expect your weight to be?
A. Two-fold
B. One-half
C. One-fourth
D. Unaffected

Answer 1

To solve this problem, first find the volume of earth before expansion and after expansion. It is given that the volume after the explosion becomes 8-times the earlier volume. So, substitute the volumes in this condition and find the radius of earth after explosion in terms of radius of earth before explosion. Then, use the formula for gravitational acceleration. Find the gravitational acceleration acting on earth before explosion and then find the same after explosion. Substitute the radius of earth after explosion in terms of radius before explosion. Then compare this equation for gravitational acceleration with the equation for gravitational acceleration before explosion. This will give the weight.

Formula used:
$V= \dfrac {4}{3}\pi {R}^{3}$
$g= \dfrac {GM}{{R}^{2}}$

Complete answer:
Let the initial volume of the earth be ${V}_{1}$
the initial radius of the earth be ${R}_{1}$
The new volume of the earth be ${V}_{2}$
the new radius of the earth be ${R}_{2}$
It is given that earth expands to make its volume eight-fold
$\Rightarrow {V}_{2}= 8{V}_{1}$ …(2)
Volume of a sphere is given by,
$V= \dfrac {4}{3}\pi {R}^{3}$
Using above equation, volume of earth before expansion is given by,
${V}_{1}= \dfrac {4}{3}\pi {{R}_{1}}^{3}$ …(2)
Similarly, volume of the earth after expansion is given by,
${V}_{2}= \dfrac {4}{3}\pi {{R}_{2}}^{3}$ ….(3)
Substituting equation. (2) in equation. (3) we get,
$\dfrac {4}{3}\pi {{R}_{2}}^{3}=8 \times \dfrac {4}{3}\pi {{R}_{1}}^{3}$
$\Rightarrow {{R}_{2}}^{3}= 8{{R}_{1}}^{3}$
Taking the cube root on both the sides we get,
${R}_{2}= 2{R}_{1}$ …(4)
We know, formula for gravitational acceleration is given by,
$g= \dfrac {GM}{{R}^{2}}$
Where, M is the mass of the earth
R is the distance from the center
Gravitational acceleration before expansion can be written as,
${g}_{0}= \dfrac {GM}{{{R}_{1}}^{2}}$ …(5)
Similarly, Gravitational acceleration after expansion can be written as,
$g= \dfrac {GM}{{{R}_{2}}^{2}}$
Substituting equation. (4) in above equation we get,
$g= \dfrac {GM}{{2{R}_{1}}^{2}}$
$\Rightarrow g= \dfrac {GM}{4{{R}_{1}}^{2}}$
Substituting equation. (5) in above equation we get,
$g= \dfrac {1}{4}{g}_{0}$
Hence, the weight of the body will be one-fourth of the initial weight.

So, the correct answer is “Option C”.

Note:
If the mass of earth gets doubled then our weight will also get doubled as the gravitational force is directly proportional to the mass of a body. But doubling the radius of earth, will decrease the weight of a body by a factor of $\dfrac {1}{4}$. This is because the gravitational force is inversely proportional to the square of the radius of earth. Students must remember these proportionalities as it makes it easier for them to answer these types of questions.