Problem 74 - Fundamentals of Physics

Question

A wheel rotates clockwise about its central axis with an angular momentum of \(600 \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}\). At time \(t=0,\) a torque of magnitude \(50 \mathrm{~N} \cdot \mathrm{m}\) is applied to the wheel to reverse the rotation. At what time \(t\) is the angular speed zero?

Step-by-Step Solution

Answer

The angular speed is zero at 12 seconds.

Step 1: Understand the Problem

We have a wheel rotating with a given initial angular momentum, and a torque is applied to it to reverse the rotation. We need to find when the angular speed becomes zero, which means the angular momentum will be zero.

Step 2: Relate Torque and Angular Momentum

Torque is the rate of change of angular momentum. This relationship is described by the equation: \ \( \tau = \frac{dL}{dt} \), where \(\tau\) is the torque and \(L\) is the angular momentum.

Step 3: Write the Equation for Change in Angular Momentum

We have \( \tau = 50 \, \text{N}\cdot\text{m} \) and initial angular momentum \( L_0 = 600 \, \text{kg}\cdot\text{m}^2/\text{s} \). We want to find the time \( t \) when the angular momentum is zero. Using \( L(t) = L_0 - \tau t \), set \( L(t) = 0 \).

Step 4: Solve for Time t

Set the equation \( 600 - 50t = 0 \) to find \( t \): 1. Rearrange to find \( t \): \ \( 50t = 600 \)2. Divide both sides by 50: \ \( t = \frac{600}{50} = 12 \) seconds.

Key Concepts

Understanding Torque

Torque is a fundamental concept in the realm of rotational motion. It represents the force that causes an object to rotate around an axis. The magnitude of the torque depends on two primary factors: the size of the applied force and the distance from the axis at which it's applied. It's a little like using a wrench where the force applied at the longer end makes it easier to turn a bolt.
In mathematical terms, torque (\(\tau\))) is given by the equation:

• \( \tau = r \times F \sin{\theta} \)

where \(r\) is the distance from the pivot, \(F\) is the force applied, and \(\theta\) is the angle between the force and the lever arm. Torque is not just about how much force you apply but also where and how you apply it, making it a significant player in rotational dynamics.

Exploring Angular Speed

Angular speed is the rate at which an object rotates or revolves around a point or axis. It tells you how quickly the angle of rotation is changing, typically measured in radians per second. Imagine a spinning CD; the faster it spins, the higher its angular speed.
Angular speed is essential in understanding the timing of movements in various applications, from the spin of planets to the cogs in machines. When an external torque affects an object, it can change not just how fast it moves, but also its direction of motion. In our example, with the applied torque, the wheel's angular speed changes until it comes to a stop, before potentially reversing its rotation.

Examining Change in Angular Momentum

Angular momentum is conserved in a closed system and is a measure of an object's rotational inertia and angular velocity. When an external force induces a change, evaluating this change in angular momentum helps us understand and predict the object’s new state of rotation.
To analyze this change, we use the relationship:

• \( \frac{dL}{dt} = \tau \)

This tells us how rapidly the angular momentum \(L\) of a system changes over time due to the applied torque \(\tau\). In our wheel problem, the initial angular momentum was \(600 \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}\), and the torque applied changes this momentum linearly to zero in 12 seconds. This calculation illustrates how torque and time interplay to alter the rotational state of a system, often utilized in engineering and physics to control motion precisely.