Log out Go to App
Go to App

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.

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 1.10 -kg bucket is tied to a rope that is wrapped around a pole mounted horizontally on friction-less bearings. The cylindrical pole has a diameter of \(0.340 \mathrm{m}\) and a mass of \(2.60 \mathrm{kg} .\) When the bucket is released from rest, how long will it take to fall to the bottom of the well, a distance of \(17.0 \mathrm{m} ?\)
A solid sphere is released from rest and allowed to roll down a board that has one end resting on the floor and is tilted at \(30^{\circ}\) with respect to the horizontal. If the sphere is released from a height of \(60 \mathrm{cm}\) above the floor, what is the sphere's speed when it reaches the lowest end of the board?
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)?
The string in a yo-yo is wound around an axle of radius \(0.500 \mathrm{cm} .\) The yo-yo has both rotational and translational motion, like a rolling object, and has mass \(0.200 \mathrm{kg}\) and outer radius \(2.00 \mathrm{cm} .\) Starting from rest, it rotates and falls a distance of \(1.00 \mathrm{m}\) (the length of the string). Assume for simplicity that the yo-yo is a uniform circular disk and that the string is thin compared to the radius of the axle. (a) What is the speed of the yo-yo when it reaches the distance of $1.00 \mathrm{m} ?$ (b) How long does it take to fall? [Hint: The transitional and rotational kinetic energies are related, but the yo-yo is not rolling on its outer radius.]
A sign is supported by a uniform horizontal boom of length \(3.00 \mathrm{m}\) and weight \(80.0 \mathrm{N} .\) A cable, inclined at an angle of \(35^{\circ}\) with the boom, is attached at a distance of \(2.38 \mathrm{m}\) from the hinge at the wall. The weight of the sign is \(120.0 \mathrm{N} .\) What is the tension in the cable and what are the horizontal and vertical forces \(F_{x}\) and \(F_{y}\) exerted on the boom by the hinge? Comment on the magnitude of \(F_{y}\).
See all solutions

Recommended explanations on Physics Textbooks

Solid State Physics

Read Explanation

Fields in Physics

Read Explanation

Particle Model of Matter

Read Explanation

Dynamics

Read Explanation

Waves Physics

Read Explanation

Famous Physicists

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