Question

If the radius of the earth were to shrink by one percent, its mass remaining the same, the acceleration due to gravity on the earth's surface would

• A. decrease
• B. remain unchanged
• C. increase
• D. be zero

Answer 1

The correct option is C

increase

Explanation for the correct option:

Option C:

Step 1:Given data:

The mass of the earth is given by =M

The radius of the earth is given by = R

We have given that the radius of the earth was to shrink by 1%.

Step 2: Expression for the acceleration due to gravity:

We know that expression for the acceleration due to gravity,
$$g={\displaystyle \frac{GM}{R^2}}$$…… (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,
$$g\propto {\displaystyle \frac{1}{R^2}}$$

Step 3: Expression of the new radius of the earth:

We have given that the radius of the earth was to shrink by 1%, therefore, we can express the new radius of the earth as,
$$R^{\prime }=R-1\mathrm{\%}R$$
$$\Rightarrow R^{\prime }=R-0.01R$$
$$\Rightarrow R^{\prime }=0.99R$$

Step 4: Expression for the new acceleration due to gravity:

We can express the new acceleration due to the gravity of the earth with this radius as,

$$g^{\prime }={\displaystyle \frac{GM}{{R^{\prime }}^2}}$$

Step 5: Calculation the new acceleration due to gravity:
Substituting $$R^{\prime }=0.99R$$ in the above equation, we get,
$$g^{\prime }={\displaystyle \frac{GM}{{\left(0.99R\right)}^2}}$$
$$\Rightarrow g^{\prime }=\left({\displaystyle \frac{1}{0.98}}\right)\left({\displaystyle \frac{GM}{R^2}}\right)$$
Using equation (1), we can write the above equation as,
$$g^{\prime }={\displaystyle \frac{g}{0.98}}$$

Step 6: Comparison of the new acceleration due to gravity:

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,
$$g^{\prime }-g={\displaystyle \frac{g}{0.98}}-g$$
$$\Rightarrow g^{\prime }-g=0.020g$$

Since the above quantity is positive, we can say that the acceleration due to the gravity of the earth is increased.

Hence, the correct answer is option (C).