A flywheel's radius is increased by 2% while keeping the mass unchanged. Find the increase in its moment of inertia (in %) about the central axis.

Explanation

To solve the problem of finding the increase in the moment of inertia of a flywheel when its radius is increased by 2% while keeping its mass unchanged, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Moment of Inertia Formula**:
The moment of inertia (I) of a flywheel about its central axis is given by the formula:
\[ I = m r^2 \]
where \( m \) is the mass and \( r \) is the radius of the flywheel.

2. **Identify the Change in Radius**:
We are given that the radius is increased by 2%. This can be expressed mathematically as:
\[ \frac{\Delta r}{r} \times 100 = 2\% \]
From this, we can deduce:
\[ \frac{\Delta r}{r} = 0.02 \]

3. **Apply the Percentage Change Formula**:
The percentage change in moment of inertia can be expressed using the formula for percentage change:
\[ \frac{\Delta I}{I} \times 100 = \frac{\Delta m}{m} \times 100 + n \cdot \frac{\Delta r}{r} \times 100 \]
where \( n \) is the power of \( r \) in the moment of inertia formula. Here, \( n = 2 \) since \( I \propto r^2 \).

4. **Substituting Known Values**:
Since the mass is unchanged, \( \Delta m = 0 \). Therefore, the equation simplifies to:
\[ \frac{\Delta I}{I} \times 100 = 2 \cdot \frac{\Delta r}{r} \times 100 \]
Substituting \( \frac{\Delta r}{r} = 0.02 \):
\[ \frac{\Delta I}{I} \times 100 = 2 \cdot 0.02 \times 100 \]

5. **Calculate the Percentage Increase**:
\[ \frac{\Delta I}{I} \times 100 = 2 \cdot 0.02 \cdot 100 = 4\% \]

6. **Conclusion**:
The increase in the moment of inertia about the central axis is:
\[ \text{Increase in moment of inertia} = 4\% \]

### Summary:
The percentage increase in the moment of inertia of the flywheel when the radius is increased by 2% while keeping the mass unchanged is **4%**.