Question

A pulley wheel \(8.14 \mathrm{~cm}\) in diameter has a \(5.63\) -m-long cord wrapped around its periphery. Starting from rest, the wheel is given an angular acceleration of \(1.47 \mathrm{rad} / \mathrm{s}^{2} .\) (a) Through what angle must the wheel turn for the cord to unwind? \((b)\) How long does it take?

Short answer

The wheel must turn through an angle of \(536 / C \times 2\pi\) radians, and this will take a time of \( \sqrt{2 \theta / \alpha} \) seconds.

Step-by-step solution

Find the Angle that the Wheel Must Turn to Allow the Cord to Unwind

Starting with the measurements of the pulley wheel, and the length of the cord, we can calculate the circumference \(C\) of the pulley using the formula \(C = \pi D\), where \(D\) is the diameter of the wheel. Since \(D = 8.14\) cm, it means \(C = \pi \times 8.14\) cm. Since the length of the cord is \(5.36\) m (or \(536\) cm), unwinding it would mean that the wheel has turned \(536 / C\) times. But since \(2\pi\) rad is equivalent to one full turn, it means that the wheel must turn through an angle of \(536 / C \times 2\pi\) rad.

Find the Time it Takes for the Wheel to Turn through this Angle

From the relationship used in angular movement \( \theta = (1/2) \alpha t^{2} \) where \( \theta \) is the angular displacement, \( \alpha \) is the angular acceleration and \( t \) is the time it took for this displacement, we can find \( t \) as \( t = \sqrt{2 \theta / \alpha}. \) Applying this formula, where \( \theta = 536 / C \times 2\pi \) rad and \( \alpha = 1.47 \) rad/s^2, will give us \( t \). Note that \( t \) will be in seconds since the units of \( \alpha \) were rad/s^2.

Key concepts

Understanding Angular Displacement

Angular displacement plays a fundamental role in understanding rotational movement and is key in many physics problems, including those involving pulley wheels. Imagine a point on the outer edge of a spinning object, such as a pulley wheel. As the object rotates, the point moves along a circular path. The angle between the starting position and final position of the point measured from the center of the circle is known as the angular displacement, denoted by the symbol \( \theta \).

In our exercise, the angular displacement corresponds to how much the pulley wheel must turn for the cord to completely unwind. To calculate this, we relate the length of the cord to the circumference of the wheel, given by \( \pi \times D \), where \( D \) is the wheel's diameter. By dividing the total length of the unwound cord by the wheel's circumference, we determine the total number of full rotations. Multiplying this number by \( 2\pi \) radians (since a full rotation corresponds to \( 2\pi \) radians) gives us the angular displacement. This concept highlights the continuous relationship between linear and angular quantities, essential in many physics applications such as gears, wheels, or any rotating systems.

Pulley Wheel Physics

Pulley wheel physics encompasses concepts from rotational motion, where pulleys serve as practical examples to demonstrate these foundational physics principles. A pulley wheel, in its essence, is a simple machine that can change the direction of a force and is often used in conjunction with ropes or cords.

When you're dealing with pulley problems like in our exercise, it's important to recognize that the pulley is a circular disc which allows us to explore relationships between angular and linear motion. In the context of our exercise, we examine how the rotational motion of the pulley converts an angular acceleration to a linear movement of the cord as it unwraps. In practical terms, angular acceleration, represented by \( \alpha \), is the rate of change of angular velocity over time. In this particular problem, the angular acceleration provided is what causes the wheel to turn and the cord to unwind from a stationary start.

This acceleration must be consistent for us to precisely use the formula \( \theta = \frac{1}{2}\alpha t^2 \), where time \( t \) can be solved for to find how long it takes for the wheel to reach the required angular displacement for the cord to be fully unwound.

Angular Movement

Angular movement refers to the motion of a body around a fixed point or axis. In physics, this type of movement is characterized by terms such as angular displacement, angular velocity, and angular acceleration. To get a firm grasp of angular movement, it's essential to understand that every point on the rotating body moves through the same angle in the same amount of time, highlighting the rigid-body nature of such systems.

For example, considering our exercise about a pulley wheel, each point on the wheel's periphery undergoes the same angular movement when the wheel rotates. This means that when we talk about the wheel's angular displacement or angular acceleration, we apply these concepts to the entire wheel. Furthermore, the time taken for the wheel to rotate through a certain angle is a critical part of understanding the dynamics of the system. The formula \( \theta = \frac{1}{2} \alpha t^2 \) beautifully illustrates this relationship, encapsulating the essential elements of angular movement: the angular displacement \( \theta \), the constant angular acceleration \( \alpha \), and the time \( t \) taken to achieve this movement.

As we solve for the time in the given exercise, we apply this fundamental relation of angular movement to find how quickly the pulley wheel must turn to unwind the cord completely given a particular angular acceleration.