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Chapter 11: Problem 13

Constant Rotational Acceleration A wheel has a constant rotational acceleration of \(3.0 \mathrm{rad} / \mathrm{s}^{2}\). During a certain \(4.0 \mathrm{~s}\) interval, it turns through an angle of 120 rad. Assuming that the wheel starts from rest, how long is it in motion at the start of this \(4.0 \mathrm{~s}\) interval?

Short Answer

Expert verified
The wheel has been in motion for 4.94 seconds at the start of the 4.0-second interval.

Step by step solution

01

Identify Known Values

Identify the given values in the problem: constant rotational acceleration \[ \boldsymbol{\alpha = 3.0 \, \text{rad/s}^2} \], interval time \[ \boldsymbol{\Delta t = 4.0 \, \text{s}} \], and angle turned \[ \boldsymbol{\Delta \theta = 120 \, \text{rad}} \]. The initial angular velocity is \[ \boldsymbol{\omega_0 = 0 \, \text{rad/s}} \] since the wheel starts from rest.
02

Use the Rotational Kinematic Equation

Apply the equation for the angle turned under constant angular acceleration: \[ \Delta \theta = \omega_{0}t + \frac{1}{2}\alpha t^{2} \]. Plug in the known values: \[ 120 \, \text{rad} = 0 \times t + \frac{1}{2} \times 3.0 \, \text{rad/s}^2 \times (t)^{2} \]
03

Solve for Time Interval

Simplify and solve for \ t: \[ 120 = \frac{1}{2} \times 3.0 \times t^{2} \ 120 = 1.5t^{2} \ t^{2} = \frac{120}{1.5} = 80 \ t = \sqrt{80} \ t = 8.94 \, \text{s} \]
04

Find the Duration

Determine the time spent in motion before the 4.0 s interval using the previously calculated total time in motion: \[ \text{Total time} = t = 8.94 \, \text{s} \] The time spent in motion before the 4.0 s interval is \[ 8.94 \, \text{s} - 4.0 \, \text{s} = 4.94 \, \text{s} \].

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Key Concepts

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

constant rotational acceleration
In this exercise, we come across the concept of constant rotational acceleration. When an object, like a wheel, rotates and its angular velocity changes at a uniform rate, we say it has a constant rotational acceleration. It means that the rate of change of the angular velocity is steady over time. In our problem, the wheel has a rotational acceleration of \(3.0 \, \text{rad/s}^2\), which means its angular velocity increases by \(3.0 \, \text{rad/s}\) every second.
rotational kinematic equation
To solve problems like the one given, we often use rotational kinematic equations. These equations relate angular displacement, initial angular velocity, angular acceleration, and time. The key equation used in solving our problem is:

\[ \Delta \theta = \omega_{0}t + \frac{1}{2}\alpha t^{2} \]

Here, \(\Delta \theta\) is the angular displacement, \( \omega_{0}\) is the initial angular velocity, \(\alpha\) is the angular acceleration, and \(t\) is the time. Because the wheel starts from rest, \(\omega_0 = 0 \, \text{rad/s}\). By substituting the known values into this equation, we can solve for the unknown parameters.
time interval calculation
In the given problem, calculating the time interval is crucial. To find out how long the wheel has been in motion before a specific 4.0-second interval, we use the rotational kinematic equation to express the angular displacement during this time. By substituting the values into our equation:

\[ 120 \, \text{rad} = 0 \times t + \frac{1}{2} \times 3.0 \, \text{rad/s}^2 \times t^2 \]

We simplify to get:

\[ 120 = 1.5t^2 \]

Solving this gives:

\[ t = \sqrt{80} = 8.94 \, \text{s} \]

Thus, the total time the wheel has been in motion is 8.94 seconds.
angular displacement
Angular displacement refers to the angle through which an object moves on a circular path. It is measured in radians and can be positive or negative depending on the direction of the rotation. In our problem, the wheel turns through an angular displacement of 120 radians during the 4.0-second interval in question. Understanding this concept helps us to relate rotational motion to linear motion, like comparing it to the distance an object might travel in a straight line. The given rotational motion scenario lets us apply and see the practical implications of angular displacement in action.

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Most popular questions from this chapter

A Pulley 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 \(F=\) \((0.50 \mathrm{~N} / \mathrm{s}) t+\left(0.30 \mathrm{~N} / \mathrm{s}^{2}\right) t^{2}\), with \(F\) in newtons and \(t\) in seconds. The pulley is initially at rest. At \(t=3.0 \mathrm{~s}\) what are (a) its rotational acceleration and (b) its rotational speed?

Two Solid Cylinders Two uniform solid cylinders, each rotating about its central (longitudinal) axis, have the same mass of \(1.25 \mathrm{~kg}\) and rotate with the same rotational speed of \(235 \mathrm{rad} / \mathrm{s}\), but they differ in radius. What is the rotational kinetic energy of (a) the smaller cylinder, of radius \(0.25 \mathrm{~m}\), and \((\mathrm{b})\) the larger cylinder, of radius \(0.75 \mathrm{~m}\) ?

A Disk Rotates A disk rotates about its central axis starting from rest and accelerates with constant rotational acceleration. At one time it is rotating at \(10 \mathrm{rev} / \mathrm{s} ; 60\) revolutions later, its rotational speed is \(15 \mathrm{rev} / \mathrm{s}\). Calculate (a) the rotational acceleration, (b) the time required to complete the 60 revolutions, (c) the time required to reach the 10 rev/s rotational speed, and (d) the number of revolutions from rest until the time the disk reaches the 10 rev/s rotational speed.

A Flywheel Has a Rotational Velocity At \(t_{1}=0\), a flywheel has a rotational velocity of \(4.7 \mathrm{rad} / \mathrm{s}\), a rotational acceleration of \(-0.25\) \(\mathrm{rad} / \mathrm{s}^{2}\), and a reference line at \(\theta_{1}=0 .\) (a) Through what maximum angle \(\theta^{\max }\) will the reference line turn in the positive direction? For what length of time will the reference line turn in the positive direction? At what times will the reference line be at (b) \(\theta=\frac{1}{2} \theta^{\max }\) and (c) \(\theta=-10.5\) rad (consider both positive and negative values of \(t\) )? (d) Graph \(\theta\) versus \(t\), and indicate the answers to \((\mathrm{a}),(\mathrm{b})\), and \((\mathrm{c})\) on the graph.

Thin Spherical Shell A thin spherical shell has a radius of \(1.90 \mathrm{~m}\). An applied torque of \(960 \mathrm{~N} \cdot \mathrm{m}\) gives the shell a rotational acceleration of \(6.20 \mathrm{rad} / \mathrm{s}^{2}\) about an axis through the center of the shell. What are (a) the rotational inertia of the shell about that axis and (b) the mass of the shell?
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