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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 \(4.0 \mathrm{rad} / \mathrm{s}^{2}\). (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 s, (b) 1800 radians.

Step by step solution

01

Identify Given Parameters

We are given the initial angular velocity \( \omega_0 = 120 \, \text{rad/s} \) and a constant angular deceleration of \( \alpha = -4.0 \, \text{rad/s}^2 \). The negative sign indicates the disk is slowing down.
02

Formula for Time to Stop

To find the time until the disk stops, use the equation \( \omega = \omega_0 + \alpha t \). Since the final angular velocity \( \omega = 0 \) when the disk stops, we can set up the equation: \( 0 = 120 + (-4.0)t \).
03

Solve for Time

Rearrange the equation \( 0 = 120 - 4t \) to solve for \( t \):\[ 4t = 120 \]\[ t = \frac{120}{4} = 30 \, \text{seconds} \]
04

Formula for Rotated Angle

To find the angle through which the disk rotates until it stops, use the formula for angular displacement \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \).
05

Calculate Rotated Angle

Substitute the known values into the formula: \( \theta = 120 \times 30 + \frac{1}{2} \times (-4.0) \times 30^2 \).This simplifies to:\[ \theta = 3600 - 1800 = 1800 \, \text{radians} \]

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Key Concepts

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

Angular Velocity
Angular velocity is a measure of how quickly an object rotates around a specific axis. It's like the speed of rotation. For objects in rotational motion, like a spinning disk, angular velocity is crucial to understand because it tells us how fast the disk is spinning.

In the context of the exercise, the disk begins with an initial angular velocity of \(120 \text{ rad/s} \). This tells us that every second, the disk rotates 120 radians. That's quite fast considering a full circle has about \(6.28\) radians!

When talking about angular velocity, it's essential to note:
  • It is represented by the symbol \(\omega\).
  • Measured in radians per second (\( \text{rad/s} \)).
  • Can be positive or negative depending on the direction of rotation.
Angular Acceleration
Angular acceleration tells us how the angular velocity of an object changes over time. It's like the rotational equivalent of linear acceleration. If a disk speeds up or slows down, it experiences angular acceleration.

Our exercise involves a disk that is slowing down. It starts at \(120 \text{ rad/s}\) and comes to a stop. This change in speed occurs due to constant angular acceleration of \(-4 \text{ rad/s}^2\), indicating deceleration.

Angular acceleration is defined by:
  • The symbol \(\alpha\).
  • Measured in radians per second squared (\( \text{rad/s}^2 \)).
  • A positive value suggests speeding up, while a negative indicates slowing down.
To calculate the time needed for the disk to stop, we use the equation: \[ \omega = \omega_0 + \alpha t \]Here, \(\omega = 0\) (since it's stopping), simplifying our problem to solve for time \(t\).
Angular Displacement
Angular displacement measures how much the object has rotated during a period of time. Think of it as the angle covered in a given time frame, expressed in radians.

In our scenario, once we determine how long it takes for the disk to halt, calculating angular displacement helps understand the total rotation during this transition. The formula used is:\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]
When calculating, substituting known values allows us to find exactly the angle through which the disk rotates before stopping. In this case, the disk rotates \(1800\) radians before coming to a complete stop.

Key points to remember about angular displacement:
  • Represents total rotation, not just end position relative to start.
  • Expressed in radians.
  • Dependent on both initial velocity and angular acceleration.

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Most popular questions from this chapter

The angular position of a point on a rotating wheel is given by \(\theta=2.0+4.0 t^{2}+2.0 t^{3},\) where \(\theta\) is in radians and \(t\) is in seconds. At \(t=0,\) what are (a) the point's angular position and (b) its angular velocity? (c) What is its angular velocity at \(t=4.0 \mathrm{~s} ?\) (d) Calculate its angular acceleration at \(t=2.0 \mathrm{~s}\). (e) Is its angular acceleration constant?

Calculate the rotational inertia of a meter stick, with mass \(0.56 \mathrm{~kg}\), about an axis perpendicular to the stick and located at the \(20 \mathrm{~cm}\) mark. (Treat the stick as a thin rod.)

The flywheel of an engine is rotating at \(25.0 \mathrm{rad} / \mathrm{s}\). When the engine is turned off, the flywheel slows at a constant rate and stops in \(20.0 \mathrm{~s}\). Calculate (a) the angular acceleration of the flywheel, (b) the angle through which the flywheel rotates in stopping, and (c) the number of revolutions made by the flywheel in stopping.

A vinyl record on a turntable rotates at \(33 \frac{1}{3}\) rev/min. (a) What is its angular speed in radians per second? What is the linear speed of a point on the record (b) \(15 \mathrm{~cm}\) and (c) \(7.4 \mathrm{~cm}\) from the turntable axis?

A tall, cylindrical chimney falls over when its base is ruptured. Treat the chimney as a thin rod of length \(55.0 \mathrm{~m}\) At the instant it makes an angle of \(35.0^{\circ}\) with the vertical as it falls, what are (a) the radial acceleration of the top, and (b) the tangential acceleration of the top. (Hint: Use energy considerations, not a torque.) (c) At what angle \(\theta\) is the tangential acceleration equal to \(g ?\)
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