If I is the moment of inertia of a solid body having `alpha`-coefficient of linear expansion then the change in I corresponding to a small change in temperature `DeltaT` is

Options

A. `alpha I DeltaT`
B. `(1)/(2)alpha I DeltaT`
C. `2 alpha I Delta T`
D. `3 alpha I DeltaT`

Answer

To solve the problem of finding the change in the moment of inertia \( I \) of a solid body with a coefficient of linear expansion \( \alpha \) corresponding to a small change in temperature \( \Delta T \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Moment of Inertia**:
The moment of inertia \( I \) of a solid body is defined as the sum of the products of the mass elements and the square of their distances from the axis of rotation. For a body with radius \( R \), we can express \( I \) in terms of \( R \).

2. **Change in Radius with Temperature**:
When the temperature of the body changes, the radius \( R \) of the body will also change due to thermal expansion. The change in radius \( \Delta R \) can be expressed using the coefficient of linear expansion \( \alpha \):
\[ \Delta R = R \alpha \Delta T \]
where \( \Delta T \) is the change in temperature.

3. **Differentiating Moment of Inertia**:
The moment of inertia for a solid body can be expressed as:
\[ I = k R^2 \]
where \( k \) is a proportionality constant. To find the change in moment of inertia \( \Delta I \), we differentiate \( I \) with respect to \( R \):
\[ dI = 2kR \, dR \]

4. **Substituting for \( dR \)**:
Now, substituting \( dR \) with \( \Delta R \):
\[ dI = 2kR \, \Delta R \]
Substituting \( \Delta R = R \alpha \Delta T \):
\[ dI = 2kR \cdot (R \alpha \Delta T) = 2kR^2 \alpha \Delta T \]

5. **Relating to Moment of Inertia**:
Since \( kR^2 = I \), we can replace \( kR^2 \) in the equation:
\[ dI = 2I \alpha \Delta T \]

6. **Final Result**:
Thus, the change in the moment of inertia \( \Delta I \) corresponding to a small change in temperature \( \Delta T \) is given by:
\[ \Delta I = 2I \alpha \Delta T \]

### Final Answer:
The change in moment of inertia \( \Delta I \) corresponding to a small change in temperature \( \Delta T \) is:
\[ \Delta I = 2I \alpha \Delta T \]