Problem 44 - Fundamentals of Physics

Question

A Texas cockroach of mass \(0.17 \mathrm{~kg}\) runs counterclockwise around the rim of a lazy Susan (a circular disk mounted on a vertical axle) that has radius \(15 \mathrm{~cm}\), rotational inertia \(5.0 \times 10^{-3} \mathrm{~kg} \cdot \mathrm{m}^{2},\) and frictionless bearings. The cockroach's speed (relative to the ground) is \(2.0 \mathrm{~m} / \mathrm{s},\) and the lazy Susan turns clockwise with angular speed \(\omega_{0}=2.8 \mathrm{rad} / \mathrm{s} .\) The cockroach finds a bread crumb on the rim and, of course, stops. (a) What is the angular speed of the lazy Susan after the cockroach stops? (b) Is mechanical energy conserved as it stops?

Step-by-Step Solution

Answer

(a) The angular speed of the lazy Susan after the cockroach stops is \( 1.6 \; \mathrm{rad/s} \). (b) No, mechanical energy is not conserved.

Step 1: Determine Angular Momentum Before

Identify the initial conditions for both the lazy Susan and the cockroach to determine the initial angular momentum. The initial angular speed of the lazy Susan is given by \( \omega_{0} = 2.8 \; \mathrm{rad/s} \). The initial angular momentum \( L_i \) is the sum of the angular momentum of the lazy Susan and the cockroach.1. Angular momentum of lazy Susan: \[ L_{ ext{Susan}} = I imes \omega_{0} \]where \( I = 5.0 \times 10^{-3} \; \mathrm{kg \cdot m^2} \).2. Angular momentum of cockroach: Since the cockroach runs around the rim, its angular momentum is \[ L_{ ext{cockroach}} = m imes r^2 \times \left( \frac{v}{r} \right) = m imes v imes r \]where \( m = 0.17 \; \mathrm{kg} \), \( v = 2.0 \; \mathrm{m/s} \), and \( r = 0.15 \; \mathrm{m} \).Thus, \[ L_{ ext{cockroach}} = 0.17 \times 2.0 \times 0.15 \]3. Total initial angular momentum: \[ L_i = I \times \omega_0 - m \times v \times r \] (Note: We subtract because the cockroach moves in the opposite direction with respect to the lazy Susan.)

Step 2: Determine Angular Momentum After

Once the cockroach stops, its angular momentum is zero. The angular momentum conserved after the cockroach stops is only for the lazy Susan.Let the new angular velocity of lazy Susan be \( \omega_f \).The final angular momentum \( L_f = I \times \omega_f \).From the conservation of angular momentum:\[ L_i = L_f \]This means:\[ I \times \omega_0 - m \times v \times r = I \times \omega_f \]

Step 3: Solve for Final Angular Speed

Rearrange the equation obtained from angular momentum conservation to solve for \( \omega_f \).\[ \omega_f = \frac{I \times \omega_0 - m \times v \times r}{I} \]Substitute the values:\[\omega_f = \frac{(5.0 \times 10^{-3}) \times 2.8 - 0.17 \times 2.0 \times 0.15}{5.0 \times 10^{-3}} \]Calculate to find \( \omega_f \).

Step 4: Check Conservation of Mechanical Energy

To see if mechanical energy is conserved, consider the kinetic energy before and after the cockroach stops.1. Initial mechanical energy: - Rotational kinetic energy of lazy Susan: \[ KE_{ ext{Susan initial}} = \frac{1}{2} \times I \times \omega_0^2 \]- Translational kinetic energy of cockroach: \[ KE_{ ext{cockroach initial}} = \frac{1}{2} \times m \times v^2 \]2. Final mechanical energy: After stopping, - Only the lazy Susan has kinetic energy: \[ KE_{ ext{Susan final}} = \frac{1}{2} \times I \times \omega_f^2 \]Compare the initial and final mechanical energies. If they are not the same, the mechanical energy is not conserved.

Key Concepts

Rotational Motion Conservation

Angular momentum forms a fundamental part of rotational motion. It mirrors how linear momentum works for translational motion, but specifically for objects that rotate around an axis. In rotational systems, the bigger the inertia or angular velocity, the greater the angular momentum. Here, we used the concept of angular momentum to analyze the cockroach and the lazy Susan.

When two objects interact, such as a cockroach running on a lazy Susan, their total angular momentum is conserved, provided no external torques affect them. This case means:

• The initial angular momentum, combining both cockroach and lazy Susan, must equal the final angular momentum.
• Given the cockroach's initial movement, its oppositional movement alters the total angular momentum of the system.
• The moment the cockroach stops, only the lazy Susan's angular momentum contributes to the final measure.

Knowing these, the conservation law helps us find the lazy Susan's angular speed post the cockroach stopping using this equation:
\[ I \times \omega_0 - m \times v \times r = I \times \omega_f \]where the left side signifies initial conditions, and the right represents final conditions.

Kinetic Energy in Rotational Motion

Kinetic energy for rotating objects, such as the lazy Susan, is fundamentally linked to their rotational inertia and angular velocity. This form of energy is known as rotational kinetic energy, expressed as:
\[ KE_{ Susan initial} = \frac{1}{2} \times I \times \omega_0^2 \]
Similarly, the cockroach running along the rim also has kinetic energy, albeit translational due to its straight-line motion. Its kinetic energy is calculated using:
\[ KE_{ cockroach initial} = \frac{1}{2} \times m \times v^2 \]

Collectively, these energies add up to the total energy of the system before the cockroach halts. When it stops, all its kinetic energy vanishes, simplifying calculations to consider only the lazy Susan’s rotational movement.
In rotational scenarios, these formulas let one evaluate how energy transitions between forms and objects. Energy is essential in grasping how mechanical systems, like our setup, maintain or lose energy through interactions.

Mechanical Energy Conservation in Rotational Systems

Understanding if mechanical energy is conserved requires a look at the energies before and after the cockroach stops moving. Though angular momentum might stay constant, energy doesn’t always follow. Here's the breakdown:

• Initial Mechanical Energy: It comprises both the rotational energy of the lazy Susan and the translational energy of the cockroach.
• Final Mechanical Energy: Post-stoppage, only the lazy Susan’s rotational energy remains.

Calculations show the energy differences, crucially indicating if energy conservation stands:
\[ KE_{ total initial} = KE_{ Susan initial} + KE_{ cockroach initial} \]
\[ KE_{ total final} = KE_{ Susan final} \]

From these, you ascertain conservation. If \( KE_{ total initial} \) differs from \( KE_{ total final} \), energy isn't conserved, usually due to transformations like heat or sound generation. Reviewing mechanical energy plays a vital role in understanding the complete dynamics within physical systems.