Problem 22 - Physics

Question

A steel bicycle wheel (without the rubber tire) is rotating freely with an angular speed of \(18.00 \mathrm{rad} / \mathrm{s}\). The temperature of the wheel changes from -100.0 to \(+300.0{ }^{\circ} \mathrm{C}\). No net external torque acts on the wheel, and the mass of the spokes is negligible. (a) Does the angular speed increase or decrease as the wheel heats up? Why? (b) What is the angular speed at the higher temperature?

Step-by-Step Solution

Answer

The angular speed decreases as the radius expands. \( \omega_f < 18.00 \mathrm{rad/s} \).

Step 1: Understanding Thermal Expansion

When the temperature of an object increases, it expands. For a circular object like a wheel, this means its radius will increase, leading to changes in its moment of inertia.

Step 2: Conservation of Angular Momentum

Since no net external torque acts on the wheel, angular momentum is conserved. This implies that the initial and final angular momenta are equal, i.e., \( I_i \omega_i = I_f \omega_f \), where \(I\) is the moment of inertia and \(\omega\) is the angular speed.

Step 3: Relation Between Moment of Inertia and Angular Speed

The moment of inertia of a ring is proportional to the square of its radius, \( I \propto R^2 \). As the wheel heats up, the radius increases, leading to an increase in the moment of inertia \( I_f > I_i \).

Step 4: Calculating New Angular Speed

Since \( I_i \omega_i = I_f \omega_f \) and \( I_f > I_i \), the angular speed \( \omega_f \) must decrease to conserve angular momentum. Thus, \( \omega_f = \omega_i \cdot \frac{I_i}{I_f} \). Given that \( I_f > I_i \), \( \omega_f < \omega_i \).

Step 5: Final Angular Speed

Although we have noted a qualitative decrease in angular speed, quantitative calculation without specific values of \(I_i\) and \(I_f\) isn't possible. However, the result is that \( \omega_f < 18.00 \mathrm{rad/s} \).

Key Concepts

Understanding Thermal Expansion Effects

When you heat an object, it expands. This is known as thermal expansion. It's most noticeable in solids, like the steel wheel of a bicycle. As the temperature increases, the steel wheel gets larger. Specifically, its radius grows.

This happens because atoms in the material move more as they warm up. They push against each other and take up more space. Consequently, as the wheel's temperature changes drastically from -100°C to +300°C, its size increases.

Understanding how thermal expansion affects materials helps in solving physics problems. When a wheel expands, it impacts its physical characteristics, particularly its moment of inertia.

Impact on Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on how mass is distributed relative to the axis of rotation. For a wheel, the farther the mass is from the axis, the higher the moment of inertia.

For a simple ring, the moment of inertia is proportional to the square of the radius, expressed mathematically as \( I \propto R^2 \). When the steel wheel expands due to thermal effects, its radius grows, and so does its moment of inertia.

An increase in the moment of inertia means the wheel will resist changes in angular motion more than before. In our exercise, the moment of inertia increases as a result of thermal expansion, affecting the angular speed as well.

Link Between Moment of Inertia and Angular Speed Change

When an object's moment of inertia changes while under the condition of conservation of angular momentum, there are consequences for its angular speed. The relation between moment of inertia \( I \) and angular speed \( \omega \) is captured by the formula \( I \omega = \text{constant} \).

In the exercise, no external torque is applied, which means the angular momentum must remain constant. As the wheel’s moment of inertia increases due to expansion, the angular speed must decrease to maintain this balance.

This phenomenon is due to the conservation of angular momentum. Because \( I_f > I_i \) implies the final moment of inertia is greater than the initial, the angular speed \( \omega_f \) must decrease, leading to \( \omega_f < 18.00 \mathrm{rad/s} \).

Physics Problem Solving Techniques

Solving physics problems involves understanding the core principles and applying them correctly. Here are a few steps to follow:

• Identify the Principle: Start by identifying the main physics principles involved. In this case, thermal expansion and conservation of angular momentum are key.
• Understand the System: Know the system you’re dealing with, such as its components and the nature of their interactions.
• Formulate Equations: Translate the physical situation into mathematical expressions. This helps predict and quantify changes.
• Calculate Using Provided Data: Use the initial conditions given in the problem, like initial angular speed, and determine outcomes based on changes.
• Check Your Assumptions: Always backtrack to check whether the assumptions made (e.g., no external torque) remain valid.

Applying these techniques ensures a structured and effective approach to problem-solving, increasing accuracy and understanding.