If the radius of the earth decreases by `10%`, the mass remaining unchanged, what will happen to the acceleration due to gravity?

Options

A. Decreases by `19%`
B. Increases by `19%`
C. Decreases by more than `19%`
D. Increases by more than `19%`

Explanation

To solve the problem of how the acceleration due to gravity changes when the radius of the Earth decreases by 10% while keeping the mass unchanged, we can follow these steps:

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

### Step 2: Define the new radius
If the radius of the Earth decreases by 10%, the new radius \( R' \) can be expressed as:
\[ R' = R - 0.1R = 0.9R \]

### Step 3: Calculate the new acceleration due to gravity
Using the new radius \( R' \), we can find the new acceleration due to gravity \( g' \):
\[ g' = \frac{G \cdot M}{(R')^2} = \frac{G \cdot M}{(0.9R)^2} \]
This simplifies to:
\[ g' = \frac{G \cdot M}{0.81R^2} \]

### Step 4: Relate the new acceleration to the original acceleration
We can relate \( g' \) to the original acceleration \( g \):
\[ g' = \frac{G \cdot M}{0.81R^2} = \frac{g}{0.81} \]

### Step 5: Calculate the factor by which \( g' \) changes
From the equation \( g' = \frac{g}{0.81} \), we can express \( g' \) in terms of \( g \):
\[ g' = 1.23g \]
This indicates that the new acceleration due to gravity is 1.23 times the original acceleration due to gravity.

### Step 6: Calculate the percentage change in acceleration due to gravity
To find the percentage change in \( g \), we calculate:
\[ \Delta g = g' - g = 1.23g - g = 0.23g \]
The percentage change is given by:
\[ \text{Percentage Change} = \frac{\Delta g}{g} \times 100\% = \frac{0.23g}{g} \times 100\% = 23\% \]

### Conclusion
Thus, the acceleration due to gravity increases by 23% when the radius of the Earth decreases by 10% while keeping the mass unchanged.

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