/*! 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 60 A hollow cylinder, a uniform sol... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 60

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.

Short Answer

Expert verified
Question: Arrange the following objects in order of increasing total kinetic energy: a hollow cylinder, a uniform solid sphere, and a uniform solid cylinder, all having the same mass and rolling on a horizontal plane without slipping at the same constant translational speed. Answer: Uniform solid sphere < Uniform solid cylinder < Hollow cylinder

Step by step solution

01

Calculate the translational kinetic energy

As all objects have the same mass and speed, we can calculate their translational kinetic energy as: \(K_t = \frac{1}{2}mv^2\)
02

Calculate the rotational kinetic energy for each object

Using each object's moment of inertia and the relationship between linear and angular velocity, we can calculate their rotational kinetic energy as follows: 1. Hollow cylinder: \(K_{r1} =\frac{1}{2}I_1\omega^2= \frac{1}{2}(mr^2)\left(\frac{v^2}{r^2}\right) = \frac{1}{2}mv^2\) 2. Uniform solid sphere: \(K_{r2} = \frac{1}{2}I_2\omega^2= \frac{1}{2}\left(\frac{2}{5}mr^2\right)\left(\frac{v^2}{r^2}\right) = \frac{1}{5}mv^2\) 3. Uniform solid cylinder: \(K_{r3} = \frac{1}{2}I_3\omega^2= \frac{1}{2}\left(\frac{1}{2}mr^2\right)\left(\frac{v^2}{r^2}\right) = \frac{1}{4}mv^2\)
03

Calculate the total kinetic energy for each object

Add the translational and rotational kinetic energy for each object: 1. Hollow cylinder: \(K_1 = K_t + K_{r1} = \frac{1}{2}mv^2 + \frac{1}{2}mv^2 = mv^2\) 2. Uniform solid sphere: \(K_2 = K_t + K_{r2} = \frac{1}{2}mv^2 + \frac{1}{5}mv^2 = \frac{7}{10}mv^2\) 3. Uniform solid cylinder: \(K_3 = K_t + K_{r3} = \frac{1}{2}mv^2 + \frac{1}{4}mv^2 = \frac{3}{4}mv^2\)
04

Order the objects from smallest to largest total kinetic energy

Comparing the total kinetic energies calculated in step 3, we find the following order: 1. Uniform solid sphere: \(K_2 = \frac{7}{10}mv^2\) 2. Uniform solid cylinder: \(K_3 = \frac{3}{4}mv^2\) 3. Hollow cylinder: \(K_1 = mv^2\) So, the total kinetic energies are ordered as: Uniform solid sphere < Uniform solid cylinder < Hollow cylinder.

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!

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 bicycle has wheels of radius \(0.32 \mathrm{m}\). Each wheel has a rotational inertia of \(0.080 \mathrm{kg} \cdot \mathrm{m}^{2}\) about its axle. The total mass of the bicycle including the wheels and the rider is 79 kg. When coasting at constant speed, what fraction of the total kinetic energy of the bicycle (including rider) is the rotational kinetic energy of the wheels?

A man is doing push-ups. He has a mass of \(68 \mathrm{kg}\) and his center of gravity is located at a horizontal distance of \(0.70 \mathrm{m}\) from his palms and \(1.00 \mathrm{m}\) from his feet. Find the forces exerted by the floor on his palms and feet.
A house painter stands \(3.0 \mathrm{m}\) above the ground on a \(5.0-\mathrm{m}\) -long ladder that leans against the wall at a point \(4.7 \mathrm{m}\) above the ground. The painter weighs \(680 \mathrm{N}\) and the ladder weighs $120 \mathrm{N} .$ Assuming no friction between the house and the upper end of the ladder, find the force of friction that the driveway exerts on the bottom of the ladder. (W) interactive: ladder; tutorial: ladder.
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 bowling ball made for a child has half the radius of an adult bowling ball. They are made of the same material (and therefore have the same mass per unit volume). By what factor is the (a) mass and (b) rotational inertia of the child's ball reduced compared with the adult ball?
See all solutions

Recommended explanations on Physics Textbooks

Physics of Motion

Read Explanation

Space Physics

Read Explanation

Modern Physics

Read Explanation

Kinematics Physics

Read Explanation

Electricity and Magnetism

Read Explanation

Mechanics and Materials

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.