Question

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

• A. Decrease by 4%
• B. Increase by 1%
• C. Increase by 4%
• D. Increase by 2%

Hint:

Since \( g \propto \frac{1}{R^2} \), small decrease in radius leads to larger increase in \( g \).

Answer 1

The Correct Option is C

Step 1: Write formula for acceleration due to gravity.
\[ g = \frac{GM}{R^2} \]

Step 2: Identify change in radius.
Radius decreases by 2%, so:
\[ R' = 0.98R \]

Step 3: Write new gravity.
\[ g' = \frac{GM}{(0.98R)^2} \]

Step 4: Simplify expression.
\[ g' = \frac{g}{(0.98)^2} \]

Step 5: Calculate denominator.
\[ (0.98)^2 \approx 0.9604 \]

Step 6: Find percentage change.
\[ g' \approx \frac{g}{0.96} \approx 1.04g \]

Step 7: Final Answer.
\[ \boxed{\text{Increase by } 4\%} \]