If the Radius of earth were shrink by 10% its mass remaining same, the acceleration due to gravity on earth.s surface would

Options

A. decrease
B. remain unchanged
C. increase
D. none

Answer

To solve the problem of how the acceleration due to gravity (g) changes when the radius of the Earth shrinks by 10% while keeping the mass constant, we can follow these steps:

### Step 1: Understand the Formula for Acceleration due to Gravity
The formula for acceleration due to gravity at the surface of a planet is given by:
\[ g = \frac{G \cdot M}{R^2} \]
where:
- \( g \) = acceleration due to gravity,
- \( G \) = universal gravitational constant,
- \( M \) = mass of the planet,
- \( R \) = radius of the planet.

### Step 2: Determine the New Radius
If the radius of the Earth shrinks by 10%, the new radius \( R' \) can be calculated as:
\[ R' = R - 0.1R = 0.9R \]
This means the new radius is 90% of the original radius.

### Step 3: Substitute the New Radius into the Formula
Now, we substitute the new radius \( R' \) into the formula for acceleration due to gravity:
\[ g' = \frac{G \cdot M}{(R')^2} = \frac{G \cdot M}{(0.9R)^2} \]

### Step 4: Simplify the Expression
Now simplify the expression:
\[ g' = \frac{G \cdot M}{(0.9^2)R^2} = \frac{G \cdot M}{0.81R^2} \]
This shows that:
\[ g' = \frac{g}{0.81} \]
where \( g \) is the original acceleration due to gravity.

### Step 5: Calculate the New Acceleration due to Gravity
Since \( \frac{1}{0.81} \) is greater than 1, it indicates that \( g' \) is greater than \( g \). Therefore, the new acceleration due to gravity will be:
\[ g' \approx 1.23g \]
This means that the acceleration due to gravity increases when the radius decreases.

### Conclusion
Thus, if the radius of the Earth shrinks by 10% while keeping the mass constant, the acceleration due to gravity at the Earth's surface will increase.

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