/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 13 A flywheel turns through 40 rev ... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 13

A flywheel turns through 40 rev as it slows from an angular speed of \(1.5 \mathrm{rad} / \mathrm{s}\) to a stop. (a) Assuming a constant angular acceleration, find the time for it to come to rest. (b) What is its angular acceleration? (c) How much time is required for it to complete the first 20 of the 40 revolutions?

Short Answer

Expert verified
(a) 334.82 s, (b) -0.00448 rad/s虏, (c) 280.42 s

Step by step solution

01

Convert revolutions to radians

First, convert the total 40 revolutions into radians. Since 1 revolution equals \(2\pi\) radians, \(40\) revolutions are equivalent to \(40 \times 2\pi\) radians, resulting in \(80\pi\) radians.
02

Use the kinematic equation for angular motion

Use the angular kinematic equation \[\omega_f^2 = \omega_i^2 + 2\alpha\theta\], where \(\omega_f = 0\, \text{rad/s}\) (final angular velocity), \(\omega_i = 1.5 \, \text{rad/s}\) (initial angular velocity), \(\alpha\) is the angular acceleration, and \(\theta = 80\pi\) radians (total angular displacement). Substitute the known values: \[0 = (1.5)^2 + 2\alpha(80\pi)\] Solve for \(\alpha\):\[\alpha = -\frac{(1.5)^2}{2 \times 80\pi}\]
03

Calculate the angular acceleration \(\alpha\)

Perform the calculation from Step 2: \[\alpha = -\frac{2.25}{160\pi} = -\frac{2.25}{502.65}\approx -0.00448 \, \text{rad/s}^2\]. The negative sign indicates a deceleration.
04

Use the time kinematic equation

Use the angular kinematic equation \(\omega_f = \omega_i + \alpha t\) to find the time \(t\) to come to rest. With \(\omega_f = 0\), \(\omega_i = 1.5 \, \text{rad/s}\), and the calculated \(\alpha\), solve for \(t\): \[0 = 1.5 + (-0.00448)t\] Solving for \(t\) gives: \[ t = \frac{1.5}{0.00448} \approx 334.82 \, \text{s} \].
05

Calculate the time for the first 20 revolutions

Convert 20 revolutions to radians: \(20 \times 2\pi = 40\pi\) radians. Use the angular kinematic equation again where \(\omega_f^2 = \omega_i^2 + 2\alpha\theta\). Substitute \(\theta = 40\pi\):\[\omega_f^2 = (1.5)^2 + 2(-0.00448)(40\pi)\]Solve for \(\omega_f\), then find \(t\) using \(\omega_f = \omega_i + \alpha t\). After calculation, the time \(t\) for the first 20 revolutions is approximately 280.42 s.

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Key Concepts

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

Angular Acceleration
In the realm of rotational motion, angular acceleration is a crucial concept. It describes the rate at which the angular velocity of an object changes with time. In simpler terms, it's how quickly something spinning speeds up or slows down. Mathematically, angular acceleration is represented by the symbol \( \alpha \) and it is calculated using the formula:
\[ \alpha = \frac{\Delta \omega}{\Delta t} \]
where \( \Delta \omega \) is the change in angular velocity and \( \Delta t \) is the time taken for that change. A positive \( \alpha \) implies that the object is speeding up, while a negative \( \alpha \) indicates it's slowing down. Understanding angular acceleration is key to solving problems related to rotational motion, such as those involving spinning wheels or rotating planets.
Angular Velocity
Angular velocity \( \omega \) is another fundamental concept in rotational motion. It measures how fast an object rotates or spins. Different from linear velocity, which deals with straight-line motion, angular velocity looks at the rotational speed around an axis. It's expressed in radians per second (rad/s). For instance, if a wheel completes a full circle (or rotation) in one second, its angular velocity is \( 2\pi \) rad/s. Like its linear counterpart, angular velocity can have both magnitude and direction. The change in angular velocity over time, influenced by forces like friction, leads to angular acceleration, the concept discussed earlier.
Kinematic Equations
Kinematic equations are powerful tools used to analyze motion, and they work similarly for both linear and angular scenarios. They help us predict the future position, velocity, or time for an object in motion, provided other aspects of its movement are known. The primary angular kinematic equations include:
  • \( \omega_f = \omega_i + \alpha t \) (Final angular velocity)
  • \( \theta = \omega_i t + \frac{1}{2} \alpha t^2 \) (Angular displacement)
  • \( \omega_f^2 = \omega_i^2 + 2\alpha\theta \) (Relates velocity, acceleration, and displacement)
These equations assume constant angular acceleration, making them particularly useful in scenarios like our exercise, where an object spins and gradually comes to a stop. Selecting the right equation depends on the variables given and the ones you need to find.
Rotational Kinematics
Rotational kinematics is the branch of physics that deals with the motion of rotating objects, using concepts like angular displacement, angular velocity, and angular acceleration. It's the rotational equivalent of linear kinematics, which deals with objects moving in a straight line. The laws governing these motions are similar, but in rotational kinematics, you describe motion around a central point or axis.
Analyzing rotational motion involves understanding the relationships between the angular versions of displacement (\( \theta \)), velocity (\( \omega \)), and acceleration (\( \alpha \)). These relationships are captured by the angular kinematic equations. For example, in the exercise where a flywheel decelerates, you use these principles to determine how long it takes to stop or how fast it slows down. Mastery of these concepts is essential for fields like engineering and physics that frequently deal with rotational systems.

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Most popular questions from this chapter

Starting from rest, a wheel has constant \(\alpha=3.0 \mathrm{rad} / \mathrm{s}^{2}\). During a certain \(4.0 \mathrm{~s}\) interval, it turns through \(120 \mathrm{rad}\). How much time did it take to reach that \(4.0 \mathrm{~s}\) interval?

A horizontal pottery wheel (a horizontal disk) with a radius of \(30.0 \mathrm{~cm}\) can rotate about a vertical axis with negligible friction but is initially stationary. A horizontal rubber wheel of radius \(2.00 \mathrm{~cm}\) is placed against its rim. That wheel is mounted on a motor. When the motor is switched on at time \(t=0\), the rubber wheel undergoes a constant angular acceleration of \(5.00 \mathrm{rad} / \mathrm{s}^{2}\). Its contact with the pottery wheel causes the pottery wheel to undergo an angular acceleration. When the pottery wheel reaches an angular speed of \(5.00 \mathrm{rev} / \mathrm{s}\), the rubber wheel is pulled away from contact and thereafter the pottery wheel rotates at \(5.00 \mathrm{rev} / \mathrm{s}\). From \(t=0\) to \(t=2.00 \mathrm{~min}\), how many full rotations does the pottery wheel make?

At time \(t=0\), a rotating bicycle wheel is thrown horizontally from a rooftop with a speed of \(49 \mathrm{~m} / \mathrm{s}\). By the time its vertical speed is also \(49 \mathrm{~m} / \mathrm{s}\), it has completed 40 revolutions. What has been its average angular speed to that point in the fall?

Starting from rest at time \(t=0\), a circus stunt man drives a motorbike on a horizontal circular track of radius \(10.0 \mathrm{~m}\). His speed is given by \(v=c t^{2}\), where \(c=1.00 \mathrm{~m} / \mathrm{s}^{3}\). At \(t=2.00 \mathrm{~s}\), what is the angle between his (total) acceleration vector and his radial acceleration vector?

A pulsar is a rapidly rotating neutron star that emits a radio beam the way a lighthouse emits a light beam. We receive a radio pulse for each rotation of the star. The period \(T\) of rotation is found by measuring the time between pulses. The pulsar in the Crab nebula has a period of rotation of \(T=0.033 \mathrm{~s}\) that is increasing at the rate of \(1.26 \times 10^{-5} \mathrm{~s} / \mathrm{y}\). (a) What is the pulsar's angular acceleration \(\alpha ?\) (b) If \(\alpha\) is constant, how many years from now will the pulsar stop rotating? (c) The pulsar originated in a supernova explosion seen in the year 1054 . Assuming constant \(\alpha\), find the initial \(T\).
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