Problem 45 - Fundamentals of Physics

Question

A man stands on a platform that is rotating (without friction) with an angular speed of 1.2 rev/s; his arms are outstretched and he holds a brick in each hand. The rotational thertia of the system consisting of the man, bricks, and platform about the central vertical axis of the platform is \(6.0 \mathrm{~kg} \cdot \mathrm{m}^{2}\). If by moving the bricks the man decreases the rotational inertia of the system to \(2.0 \mathrm{~kg} \cdot \mathrm{m}^{2},\) what are (a) the resulting angular speed of the platform and (b) the ratio of the new kinetic energy of the system to the original kinetic energy? (c) What source provided the added kinetic energy?

Step-by-Step Solution

Answer

(a) 3.6 rev/s; (b) Ratio is 3; (c) Work by the man moving the bricks.

Step 1: Understand the problem

We have a rotating system where the man can change the rotational inertia by moving the bricks. We need to calculate the new angular speed when the rotational inertia changes and then analyze the kinetic energy changes.

Step 2: Use conservation of angular momentum

Angular momentum is conserved in the absence of external torques. The initial angular momentum is the product of the initial rotational inertia and initial angular velocity, given by \( L_i = I_i \cdot \omega_i \). This equals the final angular momentum: \( L_f = I_f \cdot \omega_f \).

Step 3: Calculate final angular velocity

Set initial and final angular momentum equal: \[6.0 \cdot 1.2 = 2.0 \cdot \omega_f\]Solving for \( \omega_f \): \[\omega_f = \frac{6.0 \times 1.2}{2.0} = 3.6 \text{ rev/s}\].

Step 4: Calculate initial and final kinetic energy

Rotational kinetic energy is given by \( KE = \frac{1}{2} I \omega^2 \). Calculate both using respective rotational inertias and angular velocities:Initial: \( KE_i = \frac{1}{2} \cdot 6.0 \cdot (1.2)^2 = 4.32 \text{ J}\).Final: \( KE_f = \frac{1}{2} \cdot 2.0 \cdot (3.6)^2 = 12.96 \text{ J}\).

Step 5: Find the ratio of kinetic energies

The ratio of the final to the initial kinetic energy is given by \(\frac{KE_f}{KE_i} = \frac{12.96}{4.32} = 3\).

Step 6: Explain the source of added kinetic energy

The additional kinetic energy comes from the work done by the man when moving the bricks. This work changes the distribution of mass and thus impacts the rotational inertia.

Key Concepts

Rotational Inertia

Rotational inertia, also known as the moment of inertia, describes how difficult it is to change the rotational speed of an object. It's like mass for linear motion, but it relates to how mass is distributed with respect to the axis of rotation. Think of it as a measure of how the mass is spread out around the axis.

• When the man on the platform moves the bricks closer or further from the central axis, he changes the rotational inertia of the system.
• The closer the mass is to the axis, the less rotation inertia, making it easier to spin.
• Conversely, the further the mass is from the axis, the higher the rotational inertia, making it more difficult to change the spin.

This concept is key in understanding why the man's movements impact the platform's rotation speed. By moving the bricks inward, he decreases the rotational inertia from 6.0 kg·m² to 2.0 kg·m², increasing the angular speed.

Kinetic Energy

Kinetic energy is the energy of motion. For rotational systems, the rotational kinetic energy is given by the formula:\[ KE = \frac{1}{2} I \omega^2 \]where \( I \) is the rotational inertia and \( \omega \) is the angular velocity in revolutions per second (rev/s).

• The system initially has a kinetic energy because it is moving.
• The initial energy, with a rotational inertia of 6.0 kg·m² and angular velocity of 1.2 rev/s, calculates to 4.32 J.
• After moving the bricks and reducing the inertia to 2.0 kg·m², the kinetic energy jumps to 12.96 J, thanks to the increase in angular velocity to 3.6 rev/s.

This more significant kinetic energy in the final state compared to the initial state is crucial in demonstrating energy transfer within the system.

Work-Energy Principle

The work-energy principle explains how work done can change the energy state of a system. In this scenario, the extra kinetic energy is provided by the man's actions, as he moves the bricks.

• The man is doing work by redistributing the mass of the system, changing its rotational inertia.
• This work translates into a change in kinetic energy as the rotational inertia decreases and speed increases.
• Since energy is conserved, the work done equals the change in kinetic energy.

This principle highlights how forces, in this case from the man's muscles moving the bricks, can enhance the system's energy level without external aid.