Question

If the earth shrinks to half of its present size and its mass reduces to half of its actual mass, then the acceleration due to gravity(g) on its surface will be

• A. 4g
• B. g
• C. 2g
• D. g/2
• E. g/3

Answer 1

The Correct Option is C

Effect of Earth's Size and Mass on Gravity

If the Earth shrinks to half of its present size and its mass reduces to half of its actual mass, we need to determine the new acceleration due to gravity (g) on its surface.

Step 1: Acceleration due to Gravity Formula 

The acceleration due to gravity (g) on the surface of a planet is given by:

\( g = \frac{GM}{R^2} \)

Where:

• G is the gravitational constant.
• M is the mass of the planet.
• R is the radius of the planet.

Step 2: New Mass and Radius

The new mass (M′) is half of the original mass:

\( M' = \frac{1}{2}M \)

The new radius (R′) is half of the original radius:

\( R' = \frac{1}{2}R \)

Step 3: New Acceleration due to Gravity

The new acceleration due to gravity (g′) is:

\( g' = \frac{GM'}{R'^2} \)

\( g' = \frac{G(\frac{1}{2}M)}{(\frac{1}{2}R)^2} \)

\( g' = \frac{\frac{1}{2}GM}{\frac{1}{4}R^2} \)

\( g' = 2\frac{GM}{R^2} \)

Step 4: Relationship to Original g

Since \( g = \frac{GM}{R^2} \), we can write:

\( g' = 2g \)

Conclusion

If the Earth shrinks to half of its present size and its mass reduces to half of its actual mass, then the acceleration due to gravity (g) on its surface will be 2g.