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You are unwinding a large spool of cable. As you pull on the cable with a constant tension, what happens to the angular acceleration and angular velocity of the spool, assuming that the radius at which you are extracting the cable remains constant and there is no friction force? a) Both increase as the spool unwinds. b) Both decrease as the spool unwinds. c) Angular acceleration increases, and angular velocity decreases. d) Angular acceleration decreases, and angular velocity increases. e) It is impossible to tell.

Short Answer

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(a) Both angular acceleration and angular velocity increase (b) Both angular acceleration and angular velocity decrease (c) Angular acceleration increases, and angular velocity decreases (d) Angular acceleration decreases, and angular velocity increases Answer: (a) Both angular acceleration and angular velocity increase

Step by step solution

01

Identify the relevant equations

For this problem, we need the equation for torque (τ) and the equation relating torque and angular acceleration (α). The relevant equations are: \(τ = F \cdot r \) \(τ = I \cdot α \) where F is the force (tension) applied at a distance r from the center, and I is the moment of inertia of the spool.
02

Calculate the torque

The torque τ can be calculated using the force (tension) F and the distance r. Since F and r are both constant as mentioned in the problem, the torque is also constant: \(τ = F \cdot r \) (Constant torque)
03

Calculate the moment of inertia of the spool

We assume the spool has a cylindrical shape and uniform density. By using the equation for the moment of inertia of a cylinder, we get: \(I = \dfrac{1}{2}MR^2 \) where M is the mass of the spool and R is the radius of the spool.
04

Relate torque and angular acceleration

We can now use the equation relating torque and angular acceleration to find: \(F \cdot r = \dfrac{1}{2}MR^2 \cdot α \)
05

Evaluate the angular acceleration

Since the torque is constant, as the cable unwinds, the mass M of the spool decreases. This means the moment of inertia (I) will also decrease and, therefore, the angular acceleration α will increase: \(α \propto \dfrac{1}{M} \)
06

Evaluate the angular velocity

As the angular acceleration increases, the angular velocity ω will also increase. This can be seen from the equation related to the angular variables: \(ω_f = ω_i + α \cdot t \) Since α is increasing and the initial angular velocity ω_i is constant, the final angular velocity ω_f will also increase.
07

Choose the correct answer

From Steps 5 and 6, we see that both angular acceleration and angular velocity increase as the spool unwinds. Therefore, the correct answer is (a).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Torque
When you hear 'torque' in physics, think of it as a twist or a turn that can cause an object to rotate. In the context of a spool of cable, when you apply a constant tension to unwind the cable, you're actually applying torque to the spool. Torque is the product of the force applied and the distance from the rotational center where the force is exerted, symbolically represented as \(\tau = F \cdot r\).

Understanding torque is crucial because it’s the rotational equivalent of force for linear motion. Just as force causes an object to accelerate linearly, torque causes an object to gain angular acceleration. If either the force or the distance from the center (lever arm) is increased, the torque increases as well. For the spool, a constant pulling force at a fixed distance means a constant torque. However, it's the change in the spool's mass, as it unwinds, that affects its angular acceleration and velocity, not torque directly.
Moment of Inertia
The moment of inertia, often symbolized by \(I\), is the rotational analogue to mass in linear motion. It quantifies an object's resistance to changes in its rotation. It’s a measure of how the mass is distributed with respect to the axis of rotation; the further the mass is from the axis, the higher the moment of inertia.

For the cylindrical spool, the moment of inertia depends on its mass and radius, calculated as \(I = \frac{1}{2}MR^2\). As the spool unwinds, its mass \(M\) decreases, leading to a decrease in \(I\). Since torque (which is constant in this scenario) equals the moment of inertia times angular acceleration \(\tau = I \cdot \alpha\), a lower moment of inertia means a higher angular acceleration if the torque remains unchanged. This scenario illustrates an inverse relationship between moment of inertia and angular acceleration when torque is constant.
Angular Kinematics
Angular kinematics is all about the description of rotational motion. Key variables include angular velocity \(\omega\), which describes how fast an object spins, and angular acceleration \(\alpha\), which describes how quickly the angular velocity changes. These variables are connected through the equation \(\omega_f = \omega_i + \alpha \cdot t\), where \(\omega_i\) is the initial angular velocity, \(\omega_f\) is the final angular velocity, and \(t\) is time.

In the case of the unwinding spool, the initial pull on the cable sets it into motion with a certain angular velocity. As the mass of the spool decreases during unwinding, so does its moment of inertia, leading to an increase in angular acceleration. According to angular kinematics, an increase in angular acceleration over time will result in an increased angular velocity. Therefore, with the ongoing process of unwinding, both the angular acceleration and angular velocity of the spool increase, mirroring the interconnected dance of motion variables within angular kinematics.

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