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Chapter 10: Problem 6

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
The hollow cylinder has more kinetic energy in comparison to the solid cylinder, when they are rolling on level ground at the same speed with equal masses and radii.

Step by step solution

01

Determine the Moment of Inertia

Moment of Inertia (I) plays a significant role in rotational motion. For a solid cylinder rolling without slipping, the rotational inertia \(I\) is given by \(\frac{1}{2}mr^2\), where \(m\) is mass and \(r\) is radius of the cylinder. For a hollow cylinder, the moment of inertia \(I\) is \(mr^2\). It is worth noticing that the moment of inertia for a hollow cylinder is higher than a solid one when both have the same mass and radius.
02

Calculate Rotational Kinetic Energy

The kinetic energy (KE) of a rolling object without slippage is a sum of translational and rotational energy - \(KE = \frac{1}{2}mv^2 + \frac{1}{2}Iw^2\), Where \(v\) is the velocity and \(w\) is the angular speed. We know that velocity \(v = rw\) for rolling without slippage, so we can substitute for \(w\) to get \(KE = \frac{1}{2}mv^2 + \frac{1}{2}I(r*v)^2/mr^2\). The mass and radius are the same for both cylinders so the kinetic energy is determined by their different moments of inertia.
03

Comparing the kinetic energy

Given that all other parameters (mass, speed and radius) are equal between the two cylinders, the one with the higher moment of inertia will have greater kinetic energy. As the moment of inertia for the hollow cylinder (since \(I = mr^2\)) is greater than for solid cylinder (\(I = \frac{1}{2}mr^2\)), the hollow cylinder will have more kinetic energy.

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