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Chapter 10: Problem 45

A uniform metallic rod rotates about its perpendicular bisector with constant angular speed. If it is heated uniformly to raise its temperature slightly, (A) its speed of rotation increases. (B) its speed of rotation decreases. (C) its speed of rotation remains same. (D) its speed in increases because its moment of inertia increases.

Short Answer

Expert verified
As the temperature increases, the length of the rod expands, causing an increase in its moment of inertia. To maintain constant angular momentum, the angular speed must decrease. Therefore, the correct answer is (B) its speed of rotation decreases.

Step by step solution

01

Understand the scenario

The metallic rod is heated uniformly, which causes its temperature to rise slightly. As a result of the temperature increase, the length of the rod will also increase as metals expand when heated. Hence, the mass distribution of the rod will change which may cause changes in its moment of inertia and angular speed.
02

Analyze the moment of inertia

The moment of inertia of a uniform metallic rod rotating about its perpendicular bisector can be described by the formula: \(I = (M*L^2)/12\), where \(M\) represents the mass and \(L\) represents the length of the rod. As the temperature is increased, the length of the rod will expand, resulting in an increased moment of inertia.
03

Analyze the angular speed

Angular speed (\(\omega\)) and moment of inertia (\(I\)) are connected by the conservation of angular momentum, given by \(L = I * \omega\), where \(L\) is the angular momentum. As the moment of inertia increases due to heating, in order to conserve angular momentum, the angular speed must decrease to maintain a constant value for angular momentum.
04

Identify the correct option

As we analyzed that in order to maintain the constant angular momentum, the angular speed must decrease when the moment of inertia increases due to the rise in temperature. Thus, the correct option in this case is: (B) its speed of rotation decreases.

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