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.
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.
Reason (R): When a metal rod is heated uniformly to raise its temperature slightly, its moment of inertia increases.
Show 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.
Updated On: Jun 25, 2026
- (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
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The Correct Option is D
Solution and Explanation
Step 1: Understand the effect of heating on the rod.
When a metallic rod is heated uniformly, it expands due to thermal expansion.
Therefore, its length increases slightly.
For a uniform rod rotating about its perpendicular bisector, the moment of inertia is \[ I=\frac{1}{12}ML^2 \] where \[ M=\text{mass of the rod} \] and \[ L=\text{length of the rod} \] Since \(L\) increases on heating, \[ I \] also increases.
So, Reason (R) is true.
Step 2: Use conservation of angular momentum.
When no external torque acts on the rotating rod, angular momentum remains conserved.
Angular momentum is \[ L_{\text{angular}}=I\omega \] Since angular momentum is conserved, \[ I\omega=\text{constant} \]
Step 3: Analyze the change in angular speed.
On heating, the moment of inertia \(I\) increases.
Since \[ I\omega=\text{constant}, \] if \(I\) increases, then angular speed \(\omega\) must decrease.
Therefore, the speed of rotation does not increase. It decreases.
So, Assertion (A) is false.
Step 4: Final conclusion.
Assertion (A) is false, but Reason (R) is true.
Hence, \[ \boxed{\text{(A) is false but R is true}} \]
When a metallic rod is heated uniformly, it expands due to thermal expansion.
Therefore, its length increases slightly.
For a uniform rod rotating about its perpendicular bisector, the moment of inertia is \[ I=\frac{1}{12}ML^2 \] where \[ M=\text{mass of the rod} \] and \[ L=\text{length of the rod} \] Since \(L\) increases on heating, \[ I \] also increases.
So, Reason (R) is true.
Step 2: Use conservation of angular momentum.
When no external torque acts on the rotating rod, angular momentum remains conserved.
Angular momentum is \[ L_{\text{angular}}=I\omega \] Since angular momentum is conserved, \[ I\omega=\text{constant} \]
Step 3: Analyze the change in angular speed.
On heating, the moment of inertia \(I\) increases.
Since \[ I\omega=\text{constant}, \] if \(I\) increases, then angular speed \(\omega\) must decrease.
Therefore, the speed of rotation does not increase. It decreases.
So, Assertion (A) is false.
Step 4: Final conclusion.
Assertion (A) is false, but Reason (R) is true.
Hence, \[ \boxed{\text{(A) is false but R is true}} \]
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