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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 solid sphere has a greater rotational inertia about an axis through the center of mass than the solid cube, given that they have the same mass and the side length of the cube is equal to the diameter of the sphere.

Step by step solution

01

Formula for the sphere's rotational inertia

Start by writing down the formula for a sphere's rotational inertia: \( I_{sphere} = (2/5)mr^2 \). Here, replace \( r \) with \( L/2 \), giving \( I_{sphere} = (2/5)m(L/2)^2 \). This simplifies to \( I_{sphere} = (1/5)mL^2 \).
02

Formula for the cube's rotational inertia

Now, write down the formula for a cube's rotational inertia: \( I_{cube} = (1/6)mL^2 \).
03

Compare the rotational inertia

Finally, compare the rotational inertia of the two shapes. From the equations, we can see that \( I_{sphere} = (1/5)mL^2 \) and \( I_{cube} = (1/6)mL^2 \). As 1/5 is greater than 1/6, the sphere has the greater rotational inertia.

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Most popular questions from this chapter

Use integration to show that the rotational inertia of a thick ring of mass \(M\) and inner and outer radii \(R_{1}\) and \(R_{2}\) is given by \(\left.\frac{1}{2} M\left(R_{1}^{2}+R_{2}^{2}\right) . \text { (Hint: See Example } 10.7 .\right)\)

The local historical society has asked your assistance in writing the interpretive material for a display featuring an old steam locomotive. You have information on the torque in a flywheel but need to know the force applied by means of an attached horizontal rod. The rod joins the wheel with a flexible connection \(95 \mathrm{cm}\) from the wheel's axis. The maximum torque the rod produces on the flywheel is \(10.1 \mathrm{kN} \cdot \mathrm{m}\). What force does the rod apply?

You've got your bicycle upside-down for repairs, with its \(66-\mathrm{cm}-\) diameter wheel spinning freely at 230 rpm. The wheel's mass is 1.9 kg. concentrated mostly at the rim. You hold a wrench against the tire for \(3.1 \mathrm{s}\), applying a 2.7 - \(\mathrm{N}\) normal force. If the coefficient of friction between wrench and tire is \(0.46,\) what's the final angular speed of the wheel?

Four equal masses \(m\) are located at the comers of a square of side L. connected by essentially massless rods. Find the rotational inertia of this system about an axis (a) that coincides with one side and (b) that bisects two opposite sides.

Is it possible to apply a counterclockwise torque to an object that's rotating clockwise? If so, how will the object's motion change? If not, why not?
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