Question

(a) Prove that the change in rotational inertia \(I\) with temperature of a solid object is given by \(\Delta I=2 \alpha I \Delta T .(b)\) A thin uniform brass rod, spinning freely at 230 rev/s about an axis perpendicular to it at its center, is heated without mechanical contact until its temperature increases by \(170 \mathrm{C}^{\circ}\). Calculate the change in angular velocity.

Short answer

The change in the rotational inertia with temperature is given by the formula \(\Delta I = 2\alpha I \Delta T\). The change in angular rotation is given by \( \Delta \omega = \omega - \frac{\omega}{1 + 2 \alpha \Delta T}\). Therefore, by filling in the given values, the change in angular velocity can be calculated.

Step-by-step solution

Understand the thermal expansion principle

Thermal expansion principle states that with increasing temperature, the size of a substance also increases. In the given exercise, as the height increases, so does the moment of inertia \(I\), since it's directly proportional to the square of the distance from the axis of rotation. The coefficient of linear expansion \(\alpha\) is a measure of the rate at which the size of an object changes with a change in temperature.

Find the change in the moment of inertia with temperature

The change in rotational inertia \(\Delta I\) due to temperature \(\Delta T\) can be found by the following formula: \(\Delta I = 2\alpha I \Delta T\) where \(\alpha\) is coefficient of linear expansion of the material, \(I\) is the initial moment of inertia and \(\Delta T\) is change in temperature.

Apply the law of conservation of angular momentum

The angular momentum of the rod \(L\) is equal to moment of inertia times angular speed, thus \(L = I\omega\). According to the conservation of angular momentum, if no net torque acts on a system then the total angular momentum of the system remains constant. Hence, \(L = L'\) or \(I\omega = (I + \Delta I)\omega'\). From step 2, we found \(\Delta I = 2 \alpha I \Delta T\). Replacing \(\Delta I\) we get \(I\omega = (I + 2\alpha I \Delta T)\omega'\).

Calculate the change in angular velocity

Solving the equation from step 3, the change in angular velocity is given by \(\omega' - \omega = \Delta \omega = \omega - \frac{\omega}{1 + 2 \alpha \Delta T}\), where \(\omega'\) is the final angular velocity and \(\omega\) is the initial angular velocity.

Key concepts

Conservation of Angular Momentum

Imagine a figure skater spinning with arms extended. As they pull their arms in, they spin faster. This is an example of the conservation of angular momentum, a fundamental principle in physics that states the total angular momentum of an isolated system remains constant if no external torque is applied.

Angular momentum, represented as \( L \), is the product of rotational inertia, \( I \), and angular velocity, \( \omega \): \( L = I \cdot \omega \). When a spinning object has no external forces acting on it (no torque), such as our example with the heated brass rod from the exercise, its \( L \) stays constant. However, even if \( I \) changes due to thermal expansion, the object will adjust its \( \omega \) to maintain the same \( L \), leading to a change in its angular velocity.

Thermal Expansion Principle

Many students learn that objects expand when they are heated, but the thermal expansion principle delves into the specifics of this concept. This principle describes how the dimensions of an object change in response to temperature changes.

When an object is heated, its molecules begin to move more rapidly, causing the object to increase in size. This can cause a solid to expand in every dimension, including length, breadth, and height. In physics problems, like our heated spinning rod, we often use the coefficient of linear expansion, \( \alpha \), to predict the new size of the object after experiencing a temperature change, \( \Delta T \). The linear expansion is described by the equation \( \Delta L = \alpha L_0 \Delta T \), where \( \Delta L \) is the change in length and \( L_0 \) is the initial length.

Change in Moment of Inertia

The moment of inertia, \( I \), is a measure of an object's resistance to changes in its rotation. It depends on the mass of the object and how that mass is distributed in relation to the axis of rotation. Simply put, it's the rotational equivalent of mass in linear motion.

In our exercise, as a brass rod is heated, it expands, altering the distribution of mass and consequently, its moment of inertia. For an object expanding uniformly in all dimensions due to an increase in temperature, the moment of inertia increases according to the equation \( \Delta I = 2 \alpha I \Delta T \). Here, the factor of two comes from the fact that the moment of inertia in a rod rotating about its center depends on the square of its radius — thus, linear expansion affects it twice as much.

Coefficient of Linear Expansion

The coefficient of linear expansion, \( \alpha \), is a physical property that quantifies the degree to which a material's linear dimensions are affected by temperature changes. It's typically expressed in units of reciprocal temperature, such as \(1/^\circ C\).

When we apply heat to a material, every unit length of the material expands by an amount proportional to the temperature change and the material's coefficient of linear expansion. This is particularly important in engineering and construction because it helps predict how structures will behave under different temperature conditions. In the context of our brass rod, knowing \( \alpha \) allows us to compute both the change in the rod's physical dimensions and the consequential change in its moment of inertia, crucial for understanding how its rotation will adapt when heated.