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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
The hollow cylinder has more kinetic energy.

Step by step solution

01

Identify the types of kinetic energy

Both cylinders have two types of kinetic energy: translational (caused by the motion of the center of mass) and rotational (caused by spinning around the center of mass). More precisely, the total kinetic energy is the sum of the translational kinetic energy \(K_t\) and the rotational kinetic energy \(K_r\), represented as \(K = K_t + K_r\).
02

Find the translational kinetic energy

The translational kinetic energy for each cylinder can be defined as \(K_t = \frac{1}{2} mv^2\), where \(m\) is mass and \(v\) is velocity. Since both cylinders have the same mass and speed, their translational kinetic energies are equal.
03

Find the rotational kinetic energy

The rotational kinetic energy \(K_r\) is defined as \(\frac{1}{2} I \omega^2\), where \(I\) is the moment of inertia and \( \omega \) is the angular velocity. To compare the kinetic energies of our cylinders, we need to determine their moments of inertia. For a solid cylinder, \(I = \frac{1}{2} m r^2\), while for a hollow cylinder, \(I = m r^2\). As \( \omega \) is the same for both cylinders (since they have the same linear velocity), the hollow cylinder has a greater rotational kinetic energy due to its higher moment of inertia.
04

Combine the kinetic energy

To find the total kinetic energy, you add the translational kinetic energy and the rotational kinetic energy together. The kinetic energy of the hollow cylinder is larger due to a larger rotational kinetic energy part, despite the translational kinetic energies being equal.

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