Question

Assertion (A): When a uniform metallic rod rotating about perpendicular bisector with constant angular speed is heated uniformly to raise its temperature slightly, its speed of rotation increases.
Reason (R): When a metal rod is heated uniformly to raise its temperature slightly, its moment of inertia increases.

• (A) and (R) are true and R is correct explanation of A.
• (A) and (R) are true but (R) is not correct explanation of A.
• (A) is true, but (R) is false
• (A) is false but R is true

Hint:

When a rotating body expands due to heating, its moment of inertia increases. If no external torque acts, angular momentum \[ I\omega \] remains constant, so angular velocity decreases.

Answer 1

The Correct Option is D

Step 1: Analyze the effect of heating on moment of inertia.
Heating expands the rod, increasing its length $L$. For a uniform rod rotating about its perpendicular bisector: $I = \frac{1}{12}ML^2$. Since $L$ increases and $M$ is constant, $I$ increases. Reason (R) is true.
Step 2: Apply conservation of angular momentum.
With no external torque: $L_{\text{ang}} = I\omega = \text{constant}$.
Step 3: Determine the change in angular speed.
Since $I$ increases and $I\omega = \text{constant}$: \[ \omega_{\text{new}} = \frac{I_{\text{old}}}{I_{\text{new}}}\omega_{\text{old}} < \omega_{\text{old}} \] The angular speed decreases. Assertion (A) claims it increases, so (A) is false.
Step 4: Evaluate both statements.
(R): true. (A): false (speed decreases, not increases).
Step 5: Match to the answer choices.
The correct option is: (A) is false but (R) is true.
Step 6: State the final answer.
\[ \boxed{\text{(A) is false, (R) is true}} \]