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Chapter 10: Problem 9

A solid sphere and a solid cube have the same mass, and the side of the cube is equal to the diameter of the sphere. The cube's rotation axis is perpendicular to two of its faces. Which has greater rotational inertia about an axis through the center of mass?

Short Answer

Expert verified
The cube has a greater rotational inertia compared to the sphere when having the same mass and the axis through the center of mass.

Step by step solution

01

Define variables using provided information

Both sphere and cube have same mass \( m \), and the side of the cube \( s \) is equal to the diameter of the sphere, thus \( s = 2r \), where \( r \) is the radius of the sphere.
02

Equations for sphere and cube's rotational inertia

Use the equations for moment of inertia to find those for both the sphere and the cube. For sphere \( I_{sphere} =\frac{2}{5}mr^2 \) and for the cube it is \(I_{cube} = \frac{1}{6}ms^2 \)
03

Substitute \( s \) in terms of \( r \) in cube's rotational inertia

Substitute s in cube's rotational inertia with \( 2r \) to compare both inertias. The revised equation for cube's rotational inertia would be \( I_{cube} = \frac{1}{6}m(2r)^2 = \frac{2}{3}mr^2 \)
04

Compare the rotational inertias

Now compare the rotational inerita for the sphere \( I_{sphere} = \frac{2}{5}mr^2 \) with the rotational inertia of the cube \( I_{cube} = \frac{2}{3}mr^2 \). Since \(\frac{2}{3} > \frac{2}{5}\), the cube has greater rotational inertia for the same mass and axis through the center of mass.

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