Question

If the radius of earth shrinks by 1.5% (mass remaining same), then the value of gravitational acceleration changes by
A. 2%
B. -2%
C. 3%
D. -3%

Answer 1

In this question, the test mass and the total mass of the earth remain the same, but the radius of the earth decreases. The earth becomes more compact that indicates that the gravitational force on the test mass will increase, and hence the acceleration due to gravity will also change.

Complete step by step answer:
According to the question;
Given:
The decrease in the radius of the earth is 1.5%.
Now let us consider that the test mass used in the process is m, and the mass of the earth is M. Then,
The force of attraction on the test mass initially, ${F_1} = \dfrac{{GMm}}{{{R^2}}}$
where R is the initial radius of the earth.
The weight of the test mass is W = mg
Since the weight of the test mass is the measure of the gravitational force of attraction on the body, so we can write;

\;\;\;\;{F_1} = W\\\ \Rightarrow \dfrac{{GMm}}{{{R^2}}} = mg\\\ \Rightarrow g = \dfrac{{GM}}{{{R^2}}} \end{array}$$ ……(i) Similarly, the force of attraction on the test mass after the radius change is, $${F_2} = \dfrac{{GMm}}{{{{\left( {R'} \right)}^2}}}$$ Going by the same procedure as in the previous case, we get the new acceleration due to gravity is; $$g' = \dfrac{{GM}}{{{{\left( {R'} \right)}^2}}}$$ ……(ii) Where R’ is the reduced radius of the earth which is given by; $$\begin{array}{c} R' = R - \dfrac{{1.5}}{{100}} \times R\\\ = \dfrac{{98.5}}{{100}}R \end{array}$$ We can rewrite it as, $$\dfrac{{R'}}{R} = \dfrac{{98.5}}{{100}}$$ Taking the ratio of expression (ii) and (i), we get; $$\dfrac{{g'}}{g} = {\left( {\dfrac{R}{{R'}}} \right)^2} = {\left( {\dfrac{{100}}{{98.5}}} \right)^2} = 1.030689 \approx 1.03$$ Thus, the percentage change in acceleration is $$\begin{array}{l} = \dfrac{{g' - g}}{g} \times 100\\\ = \left( {\dfrac{{g'}}{g} - 1} \right) \times 100\\\ = \left( {1.03 - 1} \right) \times 100\% \\\ = 3\% \end{array}$$ Therefore, the change is a 3% increase in the gravitational acceleration **So, the correct answer is “Option C”.** **Additional Information:** Change in the radius of the body does not affect the gravitational constant (G), as it is a constant of the equation, which is experimentally determined and does not depend on the shape, mass, or even the distance separating the bodies. **Note:** In this question, we have to find the expression for the new and old acceleration due to gravity relating to the new and old radius of the earth. Then using these values, we can get the fractional change in the acceleration due to gravity, and at last, multiplying this fractional change with 100, we get the percentage change in the gravitational acceleration.