Problem 94 - University Physics with Modern Physics

Question

The moment of inertia of the empty turntable is \(1.5 \mathrm{~kg} \mathrm{~m}^{2}\). With a constant torque of \(2.5 \mathrm{~N} \cdot \mathrm{m},\) the turntable-person system takes \(3.0 \mathrm{~s}\) to spin from rest to an angular speed of \(1.0 \mathrm{rad} / \mathrm{s}\). What is the person's moment of inertia about an axis through her center of mass? Ignore friction in the turntable axle. (a) \(2.5 \mathrm{~kg} \cdot \mathrm{m}^{2}\) (b) \(6.0 \mathrm{~kg} \cdot \mathrm{m}^{2}\) (c) \(7.5 \mathrm{~kg} \cdot \mathrm{m}^{2}\) (d) \(9.0 \mathrm{~kg} \cdot \mathrm{m}^{2}\).

Step-by-Step Solution

Answer

The person's moment of inertia is \(6.0 \mathrm{~kg} \cdot \mathrm{m}^2\).

Step 1: Understand the Given Values

We are given: 1. Moment of inertia of the turntable, \(I_{t} = 1.5 \mathrm{~kg} \cdot \mathrm{m}^2\).2. Torque applied, \(\tau = 2.5 \mathrm{~N} \cdot \mathrm{m}\).3. Time taken to reach the angular speed, \(t = 3.0 \mathrm{~s}\).4. Final angular speed, \(\omega = 1.0 \mathrm{rad/s}\). We need to find the person's moment of inertia \(I_{p}\).

Step 2: Calculate Angular Acceleration

The relationship between torque (\(\tau\)), moment of inertia (\(I\)), and angular acceleration (\(\alpha\)) is given by Newton's second law for rotation:\[ \tau = I \times \alpha \]The total moment of inertia is the sum of the moments of inertia of the turntable and the person: \(I_{total} = I_{t} + I_{p}\).Angular acceleration can also be calculated using:\[ \alpha = \frac{\Delta \omega}{\Delta t} \]\[ \alpha = \frac{1.0 \ \mathrm{rad/s} - 0}{3.0 \, \mathrm{s}} = \frac{1}{3} \, \mathrm{rad/s^2} \]

Step 3: Solve for Total Moment of Inertia

Using the relation \( \tau = I_{total} \times \alpha \), we can solve for \(I_{total}\):\[ 2.5 \mathrm{~N} \cdot \mathrm{m} = I_{total} \times \frac{1}{3} \, \mathrm{rad/s^2} \]\[ I_{total} = 2.5 \times 3 = 7.5 \, \mathrm{kg} \cdot \mathrm{m}^2 \]

Step 4: Calculate Person's Moment of Inertia

The total moment of inertia \(I_{total}\) is the sum of \(I_{t}\) and \(I_{p}\). Thus:\[ I_{t} + I_{p} = 7.5 \, \mathrm{kg} \cdot \mathrm{m}^2 \]Substitute the known \(I_{t}\):\[ 1.5 \mathrm{~kg} \cdot \mathrm{m}^2 + I_{p} = 7.5 \, \mathrm{kg} \cdot \mathrm{m}^2 \]Solve for \(I_{p}\):\[ I_{p} = 7.5 \mathrm{~kg} \cdot \mathrm{m}^2 - 1.5 \mathrm{~kg} \cdot \mathrm{m}^2 = 6.0 \mathrm{~kg} \cdot \mathrm{m}^2 \]

Step 5: Verify the Answer

The calculated moment of inertia for the person is \(6.0 \mathrm{~kg} \cdot \mathrm{m}^2\). Re-check calculations and logical flow to ensure every step is correct and consistent.

Key Concepts

Angular Acceleration

Angular acceleration is a measure of how quickly an object's rotational speed changes. It's similar to linear acceleration but for objects that are spinning. In this exercise, the turntable-person system accelerates from rest to a certain angular speed, indicating the presence of angular acceleration.
To calculate angular acceleration (\(\alpha\)), we use the formula:

• \(\alpha = \frac{\Delta \omega}{\Delta t}\).

Here, \(\Delta \omega\) is the change in angular velocity, and \(\Delta t\) is the time over which the change occurs. Since the turntable starts from rest, the initial angular velocity is zero. The problem states it reaches \(1.0 \ \mathrm{rad/s}\) in \(3.0 \ \mathrm{s}\). Therefore, \(\alpha = \frac{1.0 - 0}{3.0} = \frac{1}{3} \ \mathrm{rad/s^2}\).Understanding angular acceleration helps us relate it to other rotational quantities, like torque.

Torque

Torque is essentially the rotational equivalent of force. It causes objects to start rotating or stop them from rotating, just as force does in linear motion. In this scenario, torque (\(\tau\)) plays a crucial role in changing the rotation of the turntable-person system.
In mathematical terms, it is defined as the product of the force applied and the lever arm, or distance from the pivot point:

• \(\tau = \text{Force} \times \text{Distance from pivot}\).

Here, we know the torque provided is \(2.5 \ \mathrm{N \cdot m}\). This torque is responsible for the angular acceleration experienced by the system.
By understanding torque, we can determine how much a force will make an object rotate. Its role in rotational dynamics is as vital as force is in linear dynamics.

Rotational Motion

Rotational motion is the motion of a body about a fixed axis or point, much like the turntable-person system. When objects rotate, different dynamics come into play compared to straightforward linear motion. This involves terms like angular velocity, angular acceleration, and torque, which define how quickly and smoothly the body spins.
In the context of this exercise, understanding rotational motion helps us see how the entire system behaves when torque is applied. The basic principle is the conservation of angular momentum, but for simplifying calculations, external torques alter the state of rotation.
Just like how displacement, velocity, and acceleration govern linear motion, angular displacement, velocity, and acceleration define rotational motion. They all work together to describe how the turntable gradually speeds up over time using the applied torque.

Newton's Second Law for Rotation

Newton's Second Law for rotation is a cornerstone of understanding rotational dynamics. It draws a parallel to Newton's Second Law for linear motion but applied to rotational systems. The law states:

• \(\tau = I \times \alpha\)

where \(\tau\) is the torque, \(I\) is the moment of inertia, and \(\alpha\) is the angular acceleration. This law explains how the rotational acceleration of an object is directly proportional to the net applied torque. Furthermore, the object's inertia, or resistance to changes in rotational motion, plays a significant role.
In this exercise, applying the law helps solve for the person's moment of inertia once the total rotational effects of the system are understood. We know the moment of inertia of the turntable alone, the torque, and the resulting angular acceleration, allowing us to isolate and determine the unknown variable, the person's inertia.