Problem 29 - Master Resource Book in JEE Main Physics

Question

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 increases because its moment of inertia increases

Step-by-Step Solution

Answer

(b) the speed of rotation decreases.

Step 1: Understand the Problem

We have a uniform rod that rotates around its perpendicular bisector with a constant angular speed. The rod is then uniformly heated, which causes it to expand thermally. We need to determine how this affects the speed of rotation.

Step 2: Analyze the Effect of Heating on the Rod

When an object is heated, it expands. For a rod, this means its length increases. Because the rod is connected at its center of mass, the increase in length will lead to a change in the moment of inertia. For a rod, the moment of inertia about an axis through its center is given by \( \frac{1}{12}mL^2 \), where \( L \) is the length.

Step 3: Consider Moment of Inertia

The moment of inertia is the resistance of an object to changes in its rotational motion. As the rod expands, its length \( L \) increases, and thus its moment of inertia \( I \) increases. Moment of inertia is directly proportional to the square of the length (\( I = \frac{1}{12}mL^2 \)).

Step 4: Apply Conservation of Angular Momentum

Angular momentum \( L \) is given by the product of moment of inertia \( I \) and angular velocity \( \omega \) (\( L = I\omega \)). Because there is no net external torque, the angular momentum of the system remains constant. That means \( I_1\omega_1 = I_2\omega_2 \), where subscripts 1 and 2 represent initial and final states.

Step 5: Analyze the Change in Angular Velocity

Since \( I_2 > I_1 \) due to the increase in \( L \) when heated, and \( L = I\omega \) must remain constant, \( \omega_2 < \omega_1 \). Therefore, the angular velocity must decrease as the moment of inertia increases.

Key Concepts

Moment of Inertia

The moment of inertia, often referred to as rotational inertia, represents how much an object resists changes to its rotational motion.
For a rotating object, it plays a role similar to mass in linear motion.
The moment of inertia (\( I \)) is determined by both the mass and the distribution of that mass relative to the axis of rotation.In the context of a rod rotating about its center, \( I \) is calculated using the formula:

• \( I = \frac{1}{12} mL^2 \)

where \( m \) is the rod's mass and \( L \) is its length.
When the rod is heated and expands, \( L \) increases. Thus, since the moment of inertia is proportional to \( L^2 \), any increase in length leads to a larger moment of inertia.This enhanced moment of inertia means the rod becomes more resistant to changes in its rotational speed.

Thermal Expansion

Thermal expansion refers to the increase in size of a physical body responding to temperature changes.
For solids, such as a metallic rod, thermal expansion typically results in a change in linear dimensions.When the uniform rod in our exercise is heated, its length extends uniformly.
This happens because the molecular vibrations increase, pushing atoms further apart. The amount by which the length changes can be measured using:

• \( \Delta L = \alpha L_0 \Delta T \)

Here, \( \alpha \) is the coefficient of linear expansion, \( L_0 \) is the initial length, and \( \Delta T \) is the change in temperature.
This extension affects the rod’s moment of inertia, impacting its rotational characteristics as the length \( L \) becomes part of the inertia formula.

Rotational Motion

Rotational motion describes objects rotating around a center or axis.
The key concepts include angular velocity, moment of inertia, and angular momentum, all interrelated. In scenarios like this exercise:

• The rod spins around its perpendicular bisector, demonstrating uniform circular motion.
• It maintains an angular momentum given by the product of its moment of inertia and angular velocity.

Angular momentum is conserved when there are no external torques acting on the system. So, even as the rod expands, the system attempts to maintain its initial angular momentum.
That means any change in moment of inertia must result in an adjustment in angular velocity (and vice versa) to conserve momentum.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates or spins and is denoted by \( \omega \).
It quantifies the rate at which the angle changes with time.From the conservation of angular momentum, we know that:

• \( L = I \omega = \text{constant} \)

If moment of inertia \( I \) increases due to the rod’s length expansion (from heating), then angular velocity \( \omega \) must decrease to keep \( L \) constant, as we have \( I_1 \omega_1 = I_2 \omega_2 \).This relationship shows a crucial aspect of rotational dynamics: adjustments made by the rotational system to maintain equilibrium between its rotational inertia and speed.
In our problem, these changes lead the heated rod to slow its rotation speed as the system adapts to its expanded form.