/*! 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 16 A Disk Rotates A disk rotates ab... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 16

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.

Short Answer

Expert verified
(a) 5/12 rad/s虏, (b) 28.8 s, (c) 48 s, (d) 240 revolutions

Step by step solution

01

- Define given quantities

Let the initial rotational speed be \( \omega_0 = 10 \, \text{rev/s} \) and the final rotational speed be \( \omega_f = 15 \, \text{rev/s} \). The number of revolutions \( N \) is 60.
02

- Convert revolutions to radians

Since \(1 \, \text{rev} = 2\pi \, \text{rad}\), we convert the rotational speeds to radians per second: \( \omega_0 = 10 \, \text{rev/s} = 10 \times 2\pi \, \text{rad/s} = 20\pi \, \text{rad/s} \), \( \omega_f = 15 \, \text{rev/s} = 15 \times 2\pi \, \text{rad/s} = 30\pi \, \text{rad/s} \). The total angular displacement is \( \theta = 60 \times 2\pi = 120\pi \, \text{rad} \).
03

- Calculate the rotational acceleration

Using the kinematic equation \( \omega_f^2 = \omega_0^2 + 2\alpha\theta \), solve for \( \alpha \): \((30\pi)^2 = (20\pi)^2 + 2 \alpha (120\pi) \). Simplify to find \( \alpha = \frac{100\pi}{240\pi} = \frac{5}{12} \, \text{rad/s}^2\).
04

- Calculate the time required to complete 60 revolutions

Using \( \omega_f = \omega_0 + \alpha t \), solve for \( t \): \( 30\pi = 20\pi + \left ( \frac{5}{12} \right )t \). Rearrange to find \( t = \frac{120 \pi}{\frac{5}{12}} \approx 28.8 \, \text{s} \).
05

- Calculate the time required to reach the 10 rev/s rotational speed

Using the same formula \( t = \frac{\omega_0 - 0}{\alpha} \: t = \frac{20 \pi}{\left(\frac{5}{12} \right)} = 48 \, \text{s} \).
06

- Calculate the number of revolutions from rest until reaching 10 rev/s

Using \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \) with \( \omega_0 = 0 \: \theta = \frac{1}{2} \left( \frac{5}{12} \right)(48^2) = 480\pi \, \text{rad} = 240 \, \text{revolutions} \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

constant rotational acceleration
In rotational kinematics, constant rotational acceleration refers to when an object鈥檚 rate of change of angular velocity remains the same over time. This is similar to constant linear acceleration but applied to rotational motion.
You can express constant rotational acceleration using the symbol \( \alpha \) (alpha).
It is measured in radians per second squared (rad/s虏). When an object starts from rest and accelerates with constant rotational acceleration, the formulas to describe its motion simplify greatly, making calculations easier.
angular displacement
Angular displacement represents the total angle through which an object has rotated during its motion. It is denoted by the symbol \( \theta \) (theta).
You measure angular displacement in radians (rad). For instance, if a disk completes a full circle, its angular displacement is \( 2\pi \) radians.
In our exercise, the disk completes 60 revolutions. To find the angular displacement, multiply the number of revolutions by \( 2\pi \) (since \( 1 \, \text{rev} = 2\pi \, \text{rad} \)): \( \theta = 60 \, \text{rev} \times 2\pi \, \text{rad/rev} = 120\pi \, \text{rad} \).
kinematic equations
Kinematic equations for rotational motion are akin to those used in linear motion but adapted for angles and angular speeds. They are essential for solving problems involving rotational kinematics. Here are a few:
  • Final angular speed: \( \omega_f = \omega_0 + \alpha t \)
  • Angular displacement: \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \)
  • Final angular speed squared: \( \omega_f^2 = \omega_0^2 + 2 \alpha \theta \)
In these equations, \( \omega_0 \) is the initial angular speed, \( \omega_f \) is the final angular speed, \( t \) is time, and \( \alpha \) is the angular acceleration.
These equations help us find quantities like time, angular speed, and angular displacement when certain variables are known.
revolutions to radians conversion
When dealing with rotational motion, it is essential to convert between revolutions and radians. This conversion is crucial because kinematic equations typically require angular measurements in radians.
Remember, one full revolution equals \( 2\pi \) radians. For example, if a disk spins at 10 revolutions per second, convert this to radians per second:

\( \omega = 10 \, rev/s \times 2\pi \, rad/rev = 20\pi \, rad/s \)
For 60 revolutions:

\( \theta = 60 \, rev \times 2\pi \, rad/rev = 120\pi \, rad \).
This conversion simplifies many calculations and ensures consistency across problems.
rotational speed
Rotational speed, also known as angular velocity, is how quickly an object rotates. It is measured in radians per second (rad/s) or revolutions per second (rev/s).
Rotational speed is represented by the symbol \( \omega \). In our exercise, the disk鈥檚 rotational speeds at different times are key to solving for other variables.
To convert from revolutions per second to radians per second, multiply by \( 2\pi \):

\( \omega = 10 \, rev/s \times 2\pi = 20\pi \, rad/s \)
And for the final speed:

\( \omega_f = 15 \, rev/s \times 2\pi = 30\pi \, rad/s \).
These values are crucial for applying kinematic equations to find acceleration, time, and displacement.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Tall Cylinder-Shaped Chimney A tall, cylinder-shaped chimney falls over when its base is ruptured. Treat the chimney as a thin rod of length \(H\), and let \(\theta\) be the angle the chimney makes with the vertical. In terms of these symbols and \(g\), express the following: (a) the rotational speed of the chimney, (b) the radial acceleration of the chimney's top, and (c) the tangential acceleration of the top. (Hint: Use energy considerations, not a torque. In part (c) recall that \(\alpha=d \omega / d t .)(\mathrm{d})\) At what angle \(\theta\) does the tangential acceleration equal \(g\) ?

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 \(0.75 \mathrm{~m}\) ?

Fixed Axis An object rotates about a fixed axis, and the rotational position of a reference line on the object is given by \(\theta=\) \((0.40 \mathrm{rad}) e^{\left(2.0 \mathrm{~s}^{-1}\right) t}\). Consider a point on the object that is \(4.0 \mathrm{~cm}\) from the axis of rotation. At \(t=0\), what are the magnitudes of the point's (a) tangential component of acceleration and (b) radial component of acceleration?

Coupled Wheels In Fig. \(11-25\). wheel \(A\) of radius \(r_{A}=10 \mathrm{~cm}\) is coupled by belt \(B\) to wheel \(C\) of radius \(r_{C}=25 \mathrm{~cm}\). The rotational speed of wheel \(A\) is increased from rest at a constant rate of \(1.6 \mathrm{rad} / \mathrm{s}^{2}\) Find the time for wheel \(C\) to reach a rotational speed of \(100 \mathrm{rev} / \mathrm{min}\),assuming the belt does not slip. (Hint: If the belt does not slip, the translational speeds at the rims of the two wheels must be equal.)

Communications \(\quad\) Satellite \(A\) communications satellite is a solid cylinder with mass \(1210 \mathrm{~kg}\), diameter \(1.21 \mathrm{~m}\), and length \(1.75 \mathrm{~m}\). Prior to launching from the shuttle cargo bay, it is set spinning at \(1.52 \mathrm{rev} / \mathrm{s}\) about the cylinder axis (Fig. \(11-27)\). Calculate the satellite's (a) rotational inertia about the rotation axis and (b) rotational kinetic energy.
See all solutions

Recommended explanations on Physics Textbooks

Fluids

Read Explanation

Classical Mechanics

Read Explanation

Electric Charge Field and Potential

Read Explanation

Force

Read Explanation

Work Energy and Power

Read Explanation

Solid State Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.