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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.

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