Find the percentage change in acceleration due to gravity on the surface of earth if the radius of the earth shrinks by 1.0%, mass remaining constant.

Explanation

To find the percentage change in acceleration due to gravity (g) on the surface of the Earth when the radius of the Earth shrinks by 1% while keeping the mass constant, we can follow these steps:

### Step 1: Understand the formula for acceleration due to gravity
The acceleration due to gravity 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
According to the problem, the radius of the Earth shrinks by 1%. This can be expressed mathematically as:
\[ \frac{\Delta R}{R} \times 100 = -1\% \]
This means:
\[ \Delta R = -0.01R \]

### Step 3: Differentiate the formula for \( g \)
To find the change in \( g \), we will differentiate the formula for \( g \). Taking the natural logarithm of both sides gives us:
\[ \ln g = \ln G + \ln M - 2 \ln R \]
Differentiating both sides with respect to \( R \) yields:
\[ \frac{\Delta g}{g} = -2 \frac{\Delta R}{R} \]

### Step 4: Substitute the change in radius
We know from Step 2 that:
\[ \frac{\Delta R}{R} = -0.01 \]
Substituting this into the differentiated equation gives:
\[ \frac{\Delta g}{g} = -2 \times (-0.01) = 0.02 \]

### Step 5: Calculate the percentage change in \( g \)
Now, to find the percentage change in \( g \):
\[ \frac{\Delta g}{g} \times 100 = 0.02 \times 100 = 2\% \]

### Step 6: Determine the nature of the change
Since the radius of the Earth is decreasing, the acceleration due to gravity will increase. Therefore, the percentage change in acceleration due to gravity is an increase of 2%.

### Final Answer
The percentage change in acceleration due to gravity on the surface of the Earth, when the radius shrinks by 1%, is an increase of **2%**.

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