Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 24

The radius of a wheel is \(0.500 \mathrm{m} .\) A rope is wound around the outer rim of the wheel. The rope is pulled with a force of magnitude $5.00 \mathrm{N},$ unwinding the rope and making the wheel spin CCW about its central axis. Ignore the mass of the rope. (a) How much rope unwinds while the wheel makes 1.00 revolution? (b) How much work is done by the rope on the wheel during this time? (c) What is the torque on the wheel due to the rope? (d) What is the angular displacement \(\Delta \theta\), in radians, of the wheel during 1.00 revolution? (e) Show that the numerical value of the work done is equal to the product \(\tau \Delta \theta\)

Short Answer

Expert verified
Answer: When the wheel makes 1.00 revolution, the length of the rope unwound is equal to the circumference of the wheel, which is \(2 \pi (0.500) m\). The work done by the rope on the wheel during this time is \(5.00 \times 2 \pi (0.500) J\).

Step by step solution

01

Part (a): Find the length of unwound rope

To find the length of the rope unwound after 1 revolution of the wheel, we can think about the circumference of the rim. For every revolution, the wheel rotates through its entire circumference, so the length of the rope that unwinds is equal to the circumference of the wheel. The formula for the circumference of a circle is \(C = 2 \pi r\), where \(C\) is the circumference and \(r\) is the radius of the circle. Given the radius \(r = 0.500\) m, the circumference can be calculated as \(C = 2 \pi (0.500)\) m.
02

Part (b): Find the work done by the rope on the wheel

To find the work done by the rope on the wheel during this time, we can use the formula \(W = Fd\), where \(W\) is the work done, \(F\) is the force applied, and \(d\) is the displacement caused by the force. As we figured out in part (a), the displacement is equal to the circumference of the wheel. Substitute the given force magnitude \(F = 5.00\) N and the displacement \(d = 2 \pi (0.500)\) m into the formula to find the work done: \(W = 5.00 \times 2 \pi (0.500)\) J.
03

Part (c): Find the torque on the wheel due to the rope

Torque, denoted by \(\tau\), is given by the formula \(\tau = Fr\), where \(F\) is the applied force and \(r\) is the radius of the wheel. Given the force \(F = 5.00\) N and the radius \(r = 0.500\) m, we can calculate the torque using the formula: \(\tau = 5.00 \times 0.500\) Nm.
04

Part (d): Find the angular displacement of the wheel during 1.00 revolution

The angular displacement of the wheel is the angle through which the wheel has rotated, measured in radians. For one complete revolution, the wheel rotates 360 degrees, or \(2 \pi\) radians. Therefore, the angular displacement \(\Delta \theta = 2\pi\) radians.
05

Part (e): Show that the work done is equal to the product of torque and angular displacement

Given the torque \(\tau = 5.00 \times 0.500\) Nm and angular displacement \(\Delta\theta = 2\pi\) radians, find their product: \(\tau\Delta\theta = (5.00\times0.500) (2\pi)\) J. Previously, we calculated the work done by the rope on the wheel, which was \(W=5.00 \times 2 \pi (0.500)\) J. Comparing the results, we can see that they are equal: \(W = \tau\Delta\theta\). This confirms that the numerical value of the work done is indeed equal to the product of torque and angular displacement.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

A solid sphere of mass 0.600 kg rolls without slipping along a horizontal surface with a transnational speed of \(5.00 \mathrm{m} / \mathrm{s} .\) It comes to an incline that makes an angle of \(30^{\circ}\) with the horizontal surface. Ignoring energy losses due to friction, to what vertical height above the horizontal surface does the sphere rise on the incline?
A flywheel of mass 182 kg has an effective radius of \(0.62 \mathrm{m}\) (assume the mass is concentrated along a circumference located at the effective radius of the flywheel). (a) How much work is done to bring this wheel from rest to a speed of 120 rpm in a time interval of 30.0 s? (b) What is the applied torque on the flywheel (assumed constant)?
Verify that the units of the rotational form of Newton's second law [Eq. (8-9)] are consistent. In other words, show that the product of a rotational inertia expressed in \(\mathrm{kg} \cdot \mathrm{m}^{2}\) and an angular acceleration expressed in \(\mathrm{rad} / \mathrm{s}^{2}\) is a torque expressed in \(\mathrm{N} \cdot \mathrm{m}\).
(a) Assume the Earth is a uniform solid sphere. Find the kinetic energy of the Earth due to its rotation about its axis. (b) Suppose we could somehow extract \(1.0 \%\) of the Earth's rotational kinetic energy to use for other purposes. By how much would that change the length of the day? (c) For how many years would \(1.0 \%\) of the Earth's rotational kinetic energy supply the world's energy usage (assume a constant \(1.0 \times 10^{21} \mathrm{J}\) per year)?
A hollow cylinder, a uniform solid sphere, and a uniform solid cylinder all have the same mass \(m .\) The three objects are rolling on a horizontal surface with identical transnational speeds \(v .\) Find their total kinetic energies in terms of \(m\) and \(v\) and order them from smallest to largest.
See all solutions

Recommended explanations on Physics Textbooks

Magnetism and Electromagnetic Induction

Read Explanation

Solid State Physics

Read Explanation

Kinematics Physics

Read Explanation

Scientific Method Physics

Read Explanation

Thermodynamics

Read Explanation

Physical Quantities And Units

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.

Sign-up for free