Question

If the radius of the earth were to shrink by 1%, its mass remaining the same, the acceleration due to gravity on the earth’s surface would:
A. Decrease by 1%
B. Remain unchanged
C. Increase by 1%
D. Increase by 2%

Answer 1

Hint: Use the expression for acceleration due to gravity to express the acceleration due to gravity when the radius of the earth is R. Determine the acceleration due to gravity when the radius is decreased by 1% in terms of the initial acceleration due to gravity. Then determine the fraction decrease in the acceleration due to gravity.

Formula used:
\[g = \dfrac{{GM}}{{{R^2}}}\]
Here, G is the universal gravitational, M is the mass of the earth and R is the radius of the earth.

Complete step by step solution:
We know that expression for the acceleration due to gravity,
\[g = \dfrac{{GM}}{{{R^2}}}\] …… (1)
Here, G is the universal gravitational, M is the mass of the earth and R is the radius of the earth.
Since we have given that the mass of the earth does not change on reducing the radius, we can write,
\[g \propto \dfrac{1}{{{R^2}}}\]

We have given that the radius of the earth were to shrink by 1%, therefore, we can express the new radius of the earth as,
\[R' = R - 1\% R\]
\[ \Rightarrow R' = R - 0.01R\]
\[ \Rightarrow R' = 0.99R\]

We can express the acceleration due to gravity of the earth with this radius as,
\[g' = \dfrac{{GM}}{{{{R'}^2}}}\]

Substituting \[R' = 0.99R\] in the above equation, we get,
\[g' = \dfrac{{GM}}{{{{\left( {0.99R} \right)}^2}}}\]
\[ \Rightarrow g' = \left( {\dfrac{1}{{0.98}}} \right)\left( {\dfrac{{GM}}{{{R^2}}}} \right)\]

Using equation (1), we can write the above equation as,
\[g' = \dfrac{g}{{0.98}}\]

To check whether the acceleration due to gravity of the earth is increased or decreased, we subtract the initial acceleration due to gravity from the new acceleration due to gravity of the earth as follows,
\[g' - g = \dfrac{g}{{0.98}} - g\]
\[ \Rightarrow g' - g = 0.020g\]

Since the above quantity is the positive, we can say that the acceleration due to gravity of the earth is increased. We can determine the percentage increase in the acceleration due to gravity as follows,
\[\dfrac{{g' - g}}{g} = 0.020 \times 100\]
\[ \Rightarrow \dfrac{{g' - g}}{g} = 2\% \]

Therefore, the acceleration due to gravity of the earth would increase by 2%.

So, the correct answer is “Option D”.

Note:
Theoretically students can answer this question by referring to the expression for acceleration due to gravity, \[g = \dfrac{{GM}}{{{R^2}}}\]. Since the radius of the earth is decreasing, the acceleration due to gravity is surely going to be increased. Also, since the acceleration due to gravity is inversely proportional to the square of the radius, the percentage increase is obviously greater than 1%.