Problem 52 - Fundamentals of Physics

Question

A cockroach of mass \(m\) lies on the rim of a uniform disk of mass \(4.00 \mathrm{~m}\) that can rotate freely about its center like a merrygo-round. Initially the cockroach and disk rotate together with an angular velocity of \(0.260 \mathrm{rad} / \mathrm{s}\). Then the cockroach walks halfway to the center of the disk. (a) What then is the angular velocity of the cockroach-disk system? (b) What is the ratio \(K / K_{0}\) of the new kinetic energy of the system to its initial kinetic energy? (c) What accounts for the change in the kinetic energy?

Step-by-Step Solution

Answer

(a) The new angular velocity is \(0.347 \, \text{rad/s}.\) (b) Calculate \(\frac{K}{K_0}\) using kinetic energy formulas. (c) Kinetic energy changes because the cockroach does work internally moving towards the center.

Step 1: Understanding the Problem

We have a disk of mass \(4.00m\) and a small mass \(m\) at the rim. They rotate together with an initial angular velocity \(\omega_0 = 0.260 \, \text{rad/s}\). When the cockroach moves halfway to the center, we need to determine the new angular velocity and the ratio of kinetic energies before and after the cockroach moves.

Step 2: Initial Moment of Inertia Calculation

The initial moment of inertia \(I_0\) consists of the moment of inertia of the disk \((I_d)\) and the cockroach \((I_c)\). For the disk, \(I_d = \frac{1}{2} \times 4.00m \times R^2 = 2mR^2\). For the cockroach located at the rim, \(I_c = mR^2\). Therefore, \(I_0 = I_d + I_c = 2mR^2 + mR^2 = 3mR^2\).

Step 3: Conservation of Angular Momentum

Since no external torques act on the system, angular momentum is conserved. Thus, \(I_0 \omega_0 = I_f \omega_f\), where \(I_f\) is the final moment of inertia and \(\omega_f\) is the final angular velocity.

Step 4: Final Moment of Inertia Calculation

As the cockroach moves halfway to the center, its distance becomes \(\frac{R}{2}\). The new moment of inertia for the cockroach is \(I'_c = m \left(\frac{R}{2}\right)^2 = \frac{mR^2}{4}\). Therefore, \(I_f = I_d + I'_c = 2mR^2 + \frac{mR^2}{4} = \frac{9mR^2}{4}\).

Step 5: Calculating New Angular Velocity

Using the conservation of angular momentum from Step 3, \(3mR^2 \times 0.260 \, \text{rad/s} = \frac{9mR^2}{4} \omega_f\). Solving for \(\omega_f\), we find:\[\omega_f = \frac{4 \times 0.260}{3} = 0.347 \, \text{rad/s}.\]

Step 6: Initial Kinetic Energy Calculation

The initial rotational kinetic energy \(K_0\) is calculated as \(K_0 = \frac{1}{2} I_0 \omega_0^2\). Substitute the known values: \(I_0 = 3mR^2\) and \(\omega_0 = 0.260 \, \text{rad/s}\), to get \(K_0 = \frac{1}{2} \times 3mR^2 \times (0.260)^2\).

Step 7: Final Kinetic Energy Calculation

The final kinetic energy \(K\) is given by \(K = \frac{1}{2} I_f \omega_f^2\). Substituting \(I_f = \frac{9mR^2}{4}\) and \(\omega_f = 0.347 \, \text{rad/s}\), we find:\[K = \frac{1}{2} \times \frac{9mR^2}{4} \times (0.347)^2.\]

Step 8: Calculate the Ratio \(\frac{K}{K_0}\)

Using the formulas for \(K\) and \(K_0\) from Steps 6 and 7, calculate the ratio:\[\frac{K}{K_0} = \frac{(\frac{9mR^2}{4})(0.347)^2}{(3mR^2)(0.260)^2}.\]

Step 9: Analyze the Change in Kinetic Energy

The increase in kinetic energy is due to the work done by the cockroach as it moves towards the center, reducing its moment of inertia and increasing angular velocity.

Key Concepts

Moment of Inertia

The moment of inertia is a measure of an object's resistance to change in its rotational motion. It is often compared to mass in linear motion.
In this exercise, the moment of inertia changes as the cockroach moves. This is crucial because it affects the system's angular velocity.
Initially, the disk and the cockroach have a combined moment of inertia, which is the sum of the disk's inertia \(2mR^2\) and the cockroach's inertia \(mR^2\), making \(I_0 = 3mR^2\).

• For the disk alone, the formula used is \(I_d = \frac{1}{2}MR^2\), where \(M\) is the disk's mass.
• The cockroach, viewed as a point mass at the rim initially, follows the formula \(I_c = mR^2\).

As the cockroach walks towards the center, its portion of the moment of inertia decreases to \(\frac{mR^2}{4}\). This reduces the total inertia to \(I_f = \frac{9mR^2}{4}\).
This change allows the object to rotate faster because its rotational resistance decreases.

Rotational Kinetic Energy

Rotational kinetic energy is the energy an object possesses due to its rotation, similar to how translational kinetic energy relates to linear motion.
It’s given by the formula \(K = \frac{1}{2}I\omega^2\), where \(I\) is the moment of inertia and \(\omega\) is the angular velocity.
The initial kinetic energy \(K_0\) for the cockroach-disk system is calculated using the total initial inertia \(3mR^2\) and the initial angular velocity 0.260 rad/s. The energy is found to be \(K_0 = \frac{1}{2} \cdot 3mR^2 \cdot (0.260)^2\).
After the cockroach moves, the final kinetic energy \(K\) is determined using the reduced inertia \(\frac{9mR^2}{4}\) and the new angular velocity 0.347 rad/s, resulting in \(K = \frac{1}{2} \cdot \frac{9mR^2}{4} \cdot (0.347)^2\). This energy relationship demonstrates how the decrease in moment of inertia allows the system to gain rotational energy.

Angular Velocity Change

Angular velocity measures how fast an object rotates about an axis. It is crucial in this problem where no external torque affects the system, meaning that angular momentum is conserved.
The initial angular velocity is \(0.260 \, \text{rad/s}\). Due to conservation of angular momentum, \(I_0\omega_0 = I_f\omega_f\), the angular velocity changes inversely with the change in moment of inertia.

• Initially, both the cockroach and disk rotate with the same velocity \(\omega_0\).
• Even though the cockroach changes its position, the overall angular momentum must remain unchanged.
• The new angular velocity \(\omega_f\) is \(0.347 \, \text{rad/s}\), which increases because \(I_f < I_0\).

This relationship highlights the intriguing aspect of rotational motion: adjustments in mass distribution can change the rate of rotation, boosting the angular velocity as measured.