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Chapter 10: Problem 7

A solid cylinder and a hollow cylinder of the same mass and radius are rolling along level ground at the same speed. Which has more kinetic energy?

Short Answer

Expert verified
From the calculations, it's clear that the hollow cylinder has more kinetic energy.

Step by step solution

01

Identifying the Kinetic Energy Formulas and Given Data

We begin with the equations for kinetic energy, which are for translation ( \(KE_{t} = \frac{1}{2}mv^{2}\) ) and for rotation ( \(KE_{r} = \frac{1}{2}I\omega^{2}\) ). Here, 'm' is the mass, 'v' is the velocity, 'I' is the moment of inertia and 'ω' is the angular velocity. We are told that the masses 'm' and the speeds 'v' are the same for and that they are cylinders, so they are rolling without slipping. So, \(\omega\) can be replaced by 'v/r' where 'r' is the radius of the cylinders. However, the moments of inertia for the two cylinders will differ.
02

Calculating Moments of Inertia

The moment of inertia for a solid cylinder is given by \(I = \frac{1}{2}mr^{2}\), and for a hollow cylinder, it's \(I = mr^{2}\). In this case, the moment of inertia is greater for the hollow cylinder.
03

Calculating Total Kinetic Energy

Substituting the moments of inertia and 'ω' into the rotational kinetic energy formula, we can calculate the total kinetic energy ( \(KE_{t} + KE_{r}\) ) for both cylinders. After simplification, we find that for solid cylinder, the kinetic energy is \( \frac{3}{4}mv^{2}\) and for hollow cylinder, it is \(mv^{2}\).

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Most popular questions from this chapter

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