Problem 45 - University Physics with Modern Physics

Question

A large wooden turntable in the shape of a flat uniform disk has a radius of 2.00 m and a total mass of 120 kg. The turntable is initially rotating at 3.00 rad/s about a vertical axis through its center. Suddenly, a 70.0-kg parachutist makes a soft landing on the turntable at a point near the outer edge. (a) Find the angular speed of the turntable after the parachutist lands. (Assume that you can treat the parachutist as a particle.) (b) Compute the kinetic energy of the system before and after the parachutist lands. Why are these kinetic energies not equal?

Step-by-Step Solution

Answer

(a) The angular speed is 1.38 rad/s. (b) The initial kinetic energy is 1080 J, and the final is 494 J; the energy difference results from non-conservative forces.

Step 1: Calculate Moment of Inertia for Turntable

The moment of inertia for a uniform disk is calculated using the formula \( I = \frac{1}{2} m r^2 \), where \( m \) is the mass of the disk and \( r \) is the radius. Substituting the given values, we have \( I_{disk} = \frac{1}{2} \times 120 \times (2.00)^2 = 240 \text{ kg} \cdot \text{m}^2 \).

Step 2: Calculate Moment of Inertia for Parachutist

The parachutist can be considered as a point mass at the edge of the disk, contributing to the total moment of inertia as \( I = m r^2 \). So, \( I_{parachutist} = 70.0 \times (2.00)^2 = 280 \text{ kg} \cdot \text{m}^2 \).

Step 3: Find Total Moment of Inertia After Landing

The total moment of inertia after the parachutist lands is the sum of the moments for the disk and the parachutist: \( I_{total} = I_{disk} + I_{parachutist} = 240 + 280 = 520 \text{ kg} \cdot \text{m}^2 \).

Step 4: Apply Conservation of Angular Momentum

Angular momentum is conserved, so \( I_{initial} \omega_{initial} = I_{total} \omega_{final} \). We can rearrange this to find the final angular speed: \( \omega_{final} = \frac{I_{initial} \omega_{initial}}{I_{total}} \). With \( \omega_{initial} = 3.00 \) rad/s and \( I_{initial} = 240 \text{ kg} \cdot \text{m}^2 \), we find \( \omega_{final} = \frac{240 \times 3.00}{520} = 1.38 \) rad/s.

Step 5: Calculate Initial Kinetic Energy

The initial rotational kinetic energy is given by \( K_i = \frac{1}{2} I_{initial} \omega_{initial}^2 \). Substituting the known values, \( K_i = \frac{1}{2} \times 240 \times (3.00)^2 = 1080 \) J.

Step 6: Calculate Final Kinetic Energy

The final rotational kinetic energy is \( K_f = \frac{1}{2} I_{total} \omega_{final}^2 \). Using the values found, \( K_f = \frac{1}{2} \times 520 \times (1.38)^2 = 494 \) J.

Step 7: Discuss Energy Difference

The difference in kinetic energy \( K_i - K_f = 1080 - 494 = 586 \) J is due to non-conservative forces (like friction) acting when the parachutist lands, converting some of the rotational kinetic energy into other forms such as heat or deformation.

Key Concepts

Moment of Inertia

The moment of inertia, often represented by the symbol \( I \), plays a crucial role in rotational dynamics, much like mass in linear motion. For any rotating body, moment of inertia provides a measure of how much resistance it offers against changes in its rotational motion.
To visualize this, think of a spinning figure skater. As they extend their arms, their moment of inertia increases, making it harder for them to spin quickly. Bringing their arms in tight decreases their moment of inertia, allowing for a faster spin. This principle helps explain why the parachutist affects the turntable's angular speed.
Calculating the moment of inertia depends on the shape and mass distribution of an object. In our case, a disk's moment is given by \( I = \frac{1}{2} m r^2 \), where \( m \) is mass, and \( r \) is its radius. For the parachutist, who acts as a point mass at a distance \( r \), we use \( I = m r^2 \). Together, these principles determine how the addition of the parachutist alters the system's overall inertia.

Rotational Kinetic Energy

Just as objects in linear motion have kinetic energy, rotating objects possess rotational kinetic energy. This type of energy describes how much work an object can perform due to its spin.
Rotational kinetic energy is represented by the formula \( K = \frac{1}{2} I \omega^2 \), where \( I \) is the moment of inertia and \( \omega \) is the angular velocity. A rotating disk like our turntable has energy due to its spin, even before the parachutist lands.
When the parachutist joins the turntable, the system's rotational kinetic energy changes. Initially, the energy was high, but after the parachutist lands, the energy decreases. This difference arises because the energy must now accommodate the added mass and its effect on rotational speed, an excellent demonstration of energy conservation principles.

Non-conservative Forces

Non-conservative forces, such as friction and air resistance, are forces where energy is not conserved in mechanical form. These contrast with conservative forces, where energy can be perfectly converted between kinetic and potential forms.
In the parachutist-landing scenario, the energy loss, represented by the difference in initial and final kinetic energies, results from these non-conservative forces at play. As the parachutist lands, some of the mechanical energy transforms into other forms, such as heat or sound due to frictional interactions.
This transformation demonstrates why initial and final kinetic energies differ. Understanding non-conservative forces is crucial for analyzing real-world systems where energy is never perfectly conserved in one form. It's an essential concept for bridging theoretical physics and practical applications in everyday life.