Calculate the percentage change in acceleration due to gravity if the radius of earth shrinks by 3% keeping mass unchanged.

Answer

To solve the problem of calculating the percentage change in acceleration due to gravity when the radius of the Earth shrinks by 3% 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 universal gravitational constant,
- \( M \) is the mass of the Earth,
- \( R \) is the radius of the Earth.

### Step 2: Identify the change in radius
The problem states that the radius of the Earth shrinks by 3%. This can be expressed as:
\[ \Delta R = -0.03R \]
This indicates a decrease in radius.

### Step 3: Calculate the percentage change in radius
The percentage change in radius is given as:
\[ \frac{\Delta R}{R} \times 100 = -3\% \]

### Step 4: Use logarithmic differentiation
To find the percentage change in \( g \), we can take the natural logarithm of the equation for \( g \):
\[ \ln g = \ln G + \ln M - 2 \ln R \]
Differentiating both sides gives:
\[ \frac{\Delta g}{g} = -2 \frac{\Delta R}{R} \]

### Step 5: Substitute the percentage change in radius
From the previous step, we know:
\[ \frac{\Delta R}{R} \times 100 = -3\% \]
Thus:
\[ \frac{\Delta R}{R} = -0.03 \]
Substituting this into the equation for \( \Delta g \):
\[ \frac{\Delta g}{g} = -2 \times (-0.03) = 0.06 \]

### Step 6: Convert to percentage
To find the percentage change in \( g \):
\[ \frac{\Delta g}{g} \times 100 = 0.06 \times 100 = 6\% \]

### Conclusion
The percentage change in acceleration due to gravity when the radius of the Earth shrinks by 3% (with mass unchanged) is:
\[ \text{Percentage change in } g = 6\% \]