A uniform metallic rod rotates about its perpendicular bisector with constant angualr speed. If it is heated uniformly to raise its temperature slightly, then a) its speed of rotation increases b) its speed of rotation decreases c) its speed of rotation remains same d) its speed increases because its moment of inertia increases

Options

A. its speed of rotation increases
B. its speed of rotation decreases
C. its speed of rotation remains same
D. its speed increases because its moment of inertia increases

Explanation

To solve the question, we need to analyze the effects of heating a uniform metallic rod that is rotating about its perpendicular bisector. Here are the steps to arrive at the correct answer:

### Step 1: Understand the System
The rod is rotating about its perpendicular bisector with a constant angular speed. When an object rotates, it has a certain angular momentum, which is given by the product of its moment of inertia (I) and its angular velocity (ω):
\[ L = I \cdot \omega \]

### Step 2: Effect of Heating on the Rod
When the rod is heated uniformly, it expands. This expansion increases the length of the rod, which in turn affects its moment of inertia. The moment of inertia for a uniform rod rotating about its center is given by:
\[ I = \frac{1}{12} m L^2 \]
where \( m \) is the mass and \( L \) is the length of the rod. As the length \( L \) increases due to heating, the moment of inertia \( I \) also increases.

### Step 3: Conservation of Angular Momentum
In a closed system where no external torques are acting, angular momentum is conserved. Therefore, if the moment of inertia increases, the angular velocity must change to keep the angular momentum constant:
\[ L = I \cdot \omega = \text{constant} \]

### Step 4: Relationship Between Moment of Inertia and Angular Velocity
Since \( L \) is constant and \( I \) increases, we can express this relationship as:
\[ I_1 \cdot \omega_1 = I_2 \cdot \omega_2 \]
Where:
- \( I_1 \) and \( \omega_1 \) are the initial moment of inertia and angular velocity.
- \( I_2 \) and \( \omega_2 \) are the moment of inertia and angular velocity after heating.

Given that \( I_2 > I_1 \), it follows that:
\[ \omega_2 < \omega_1 \]
This means that the angular velocity (speed of rotation) decreases.

### Step 5: Conclusion
Therefore, the correct answer is:
**b) its speed of rotation decreases.**