The diameter of a flywheel is increased by 1% . Increase in its moment of interia about the central axis is

Options

A. `1%`
B. `0.5%`
C. `2%`
D. `4%`

Answer

To find the increase in the moment of inertia of a flywheel when its diameter is increased by 1%, 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:
\[ I = \frac{1}{2} m r^2 \]
where \( m \) is the mass and \( r \) is the radius of the flywheel.

2. **Relate Diameter to Radius**:
The diameter \( d \) is related to the radius \( r \) by the equation:
\[ d = 2r \quad \Rightarrow \quad r = \frac{d}{2} \]
Thus, we can also express the moment of inertia in terms of diameter:
\[ I = \frac{1}{8} m d^2 \]

3. **Determine the Change in Diameter**:
Given that the diameter is increased by 1%, we can express this mathematically:
\[ \Delta d = 0.01d \]
Therefore, the new diameter \( d' \) becomes:
\[ d' = d + \Delta d = d + 0.01d = 1.01d \]

4. **Calculate the New Moment of Inertia**:
The new moment of inertia \( I' \) with the new diameter \( d' \) is:
\[ I' = \frac{1}{8} m (d')^2 = \frac{1}{8} m (1.01d)^2 \]
Expanding this gives:
\[ I' = \frac{1}{8} m (1.0201d^2) = 1.0201 \cdot \frac{1}{8} m d^2 = 1.0201 I \]

5. **Calculate the Change in Moment of Inertia**:
The change in moment of inertia \( \Delta I \) is:
\[ \Delta I = I' - I = 1.0201 I - I = 0.0201 I \]

6. **Calculate the Percentage Increase**:
The percentage increase in moment of inertia is given by:
\[ \text{Percentage Increase} = \left( \frac{\Delta I}{I} \right) \times 100 = \left( \frac{0.0201 I}{I} \right) \times 100 = 2.01\% \]
For practical purposes, we can round this to approximately 2%.

### Final Answer:
The increase in the moment of inertia about the central axis is approximately **2%**.

---