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Chapter 10: Problem 11

A disk, initially rotating at \(120 \mathrm{rad} / \mathrm{s}\), is slowed down with a constant angular acceleration of magnitude \(4.0 \mathrm{rad} / \mathrm{s}^{2} .(\mathrm{a})\) How much time does the disk take to stop? (b) Through what angle does the disk rotate during that time?

Short Answer

Expert verified
(a) 30 seconds, (b) 1800 radians

Step by step solution

01

Identify the Given Data and the Unknowns

We start by extracting the given data from the problem. We know the initial angular velocity \( \omega_i \) is \( 120 \ \mathrm{rad/s} \), the final angular velocity \( \omega_f \) is \( 0 \ \mathrm{rad/s} \) since the disk stops, and the angular acceleration \( \alpha \) is \( -4.0 \ \mathrm{rad/s^2} \). We need to find the time \( t \) it takes for the disk to stop.
02

Use Angular Kinematics to Solve for Time

We use the kinematic equation for angular motion: \[\omega_f = \omega_i + \alpha t\]Plugging in the known values,\[0 = 120 \ \mathrm{rad/s} + (-4 \ \mathrm{rad/s^2}) \times t\]This simplifies to \[120 = 4t\]Solving for \( t \), we get:\[t = \frac{120}{4} = 30 \ \mathrm{seconds}\]
03

Calculate the Angle Rotated During Stopping

We use another kinematic equation for angular displacement \( \theta \): \[\theta = \omega_i t + \frac{1}{2} \alpha t^2\]Substituting the known values:\[\theta = 120 \times 30 + \frac{1}{2} \times (-4) \times (30)^2\]This becomes:\[\theta = 3600 - 1800 = 1800 \ \mathrm{radians}\]

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Key Concepts

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

Angular Velocity
Angular velocity describes how quickly an object rotates around a fixed axis. Think of it as telling you how fast a disc spins. It's often measured in radians per second (rad/s), a unit that hints at both circular motion and time. In this exercise, the initial angular velocity of the disc is given as
  • \( \omega_i = 120 \ \textrm{rad/s} \)
This means the disc spins quite swiftly at the beginning.

Understanding angular velocity helps us visualize the rapidity of rotational or spinning motions, whether it's a disc, a wheel, or even celestial bodies.
Angular Acceleration
Angular acceleration measures how the angular velocity of an object changes over time. It tells us how quickly or slowly the spinning rate increases or decreases. Its unit is radians per second squared (rad/s虏).
  • In our problem, the angular acceleration is constant at \( \alpha = -4.0 \ \textrm{rad/s}^2 \), meaning a negative sign indicates the disc slows down or decelerates.
Angular acceleration can be thought of as similar to stepping on the brakes of a car. The more you press, the faster you stop! Here, it quantifies how fast the stopping happens.
Angular Displacement
Angular displacement refers to the angle through which an object rotates during a specified period of time. It provides the rotational equivalent of linear distance. Just as you can measure how far you walk, angular displacement tells us how much you've 'turned'.
  • In this exercise, the angular displacement during deceleration is calculated as \( \theta = 1800 \ \textrm{radians} \).
This large number reflects a significant change in angular position, meaning that the disc turns many times before it comes to a stop!
Kinematic Equations
The kinematic equations for angular motion are similar to those for linear motion but adapted for rotations. They include variables like angular velocity, acceleration, and displacement all of which are crucial for solving rotational motion problems.Key equations include:
  • \( \omega_f = \omega_i + \alpha t \): Connects final and initial angular velocities with time and angular acceleration.
  • \( \theta = \omega_i t + \frac{1}{2} \alpha t^2 \): Allows for calculation of angular displacement with given values.
These equations provide a roadmap for anyone trying to decipher how objects rotate over time. By understanding how to use these equations, you can predict and calculate rotational behaviors efficiently.

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Most popular questions from this chapter

(a) Show that the rotational inertia of a solid cylinder of mass \(M\) and radius \(R\) about its central axis is equal to the rotational inertia of a thin hoop of mass \(M\) and radius \(R / \sqrt{2}\) about its central axis. (b) Show that the rotational inertia \(I\) of any given body of mass \(M\) about any given axis is equal to the rotational inertia of an equivalent hoop about that axis, if the hoop has the same mass \(M\) and a radius \(k\) given by $$ k=\sqrt{\frac{I}{M}} . $$ The radius \(k\) of the equivalent hoop is called the radius of gyration of the given body.

A pulley, with a rotational inertia of \(1.0 \times 10^{-3} \mathrm{~kg} \cdot \mathrm{m}^{2}\) about its axle and a radius of \(10 \mathrm{~cm}\), is acted on by a force applied tangentially at its rim. The force magnitude varies in time as \(F=0.50 t+0.30 t^{2}\), with \(F\) in newtons and \(t\) in seconds. The pulley is initially at rest. At \(t=3.0 \mathrm{~s}\) what are its (a) angular acceleration and (b) angular speed?

A disk rotates at constant angular acceleration, from angular position \(\theta_{1}=10.0 \mathrm{rad}\) to angular position \(\theta_{2}=70.0 \mathrm{rad}\) in \(6.00 \mathrm{~s}\). Its angular velocity at \(\theta_{2}\) is \(15.0 \mathrm{rad} / \mathrm{s}\). (a) What was its angular velocity at \(\theta_{1} ?\) (b) What is the angular acceleration? (c) At what angular position was the disk initially at rest? (d) Graph \(\theta\) versus time \(t\) and angular speed \(\omega\) versus \(t\) for the disk, from the beginning of the motion \((\operatorname{let} t=0\) then \()\).

Between 1911 and 1990 , the top of the leaning bell tower at Pisa, Italy, moved toward the south at an average rate of \(1.2 \mathrm{~mm} / \mathrm{y}\). The tower is \(55 \mathrm{~m}\) tall. In radians per second, what is the average angular speed of the tower's top about its base?

A gyroscope flywheel of radius \(2.83 \mathrm{~cm}\) is accelerated from rest at \(14.2 \mathrm{rad} / \mathrm{s}^{2}\) until its angular speed is \(2760 \mathrm{rev} / \mathrm{min} .\) (a) What is the tangential acceleration of a point on the rim of the flywheel during this spin-up process? (b) What is the radial acceleration of this point when the flywheel is spinning at full speed? (c) Through what distance does a point on the rim move during the spin-up?
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