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Chapter 10: Problem 7

Consider a cylinder and a hollow cylinder, rotating about an axis going through their centers of mass. If both objects have the same mass and the same radius, which object will have the larger moment of inertia? a) The moment of inertia will be the same for both objects. b) The solid cylinder will have the larger moment of inertia because its mass is uniformly distributed. c) The hollow cylinder will have the larger moment of inertia because its mass is located away from the axis of rotation.

Short Answer

Expert verified
Explain your answer. Answer: The hollow cylinder will have a larger moment of inertia because its mass is located away from the axis of rotation. The moment of inertia of the solid cylinder is half the moment of inertia of the hollow cylinder.

Step by step solution

01

Recall the moment of inertia formulas for solid and hollow cylinders

To compare the moment of inertia of both objects, we need to recall their formulas. The moment of inertia (I) of a solid cylinder is given by: \[I_\text{solid} = \frac{1}{2}MR^2,\] where M is the mass of the cylinder and R is its radius. For a hollow cylinder, the moment of inertia is given by: \[I_\text{hollow} = MR^2.\]
02

Compare the moment of inertia formulas

Now, we can compare the moments of inertia of the solid and hollow cylinders: \[\frac{I_\text{solid}}{I_\text{hollow}} = \frac{\frac{1}{2}MR^2}{MR^2} = \frac{1}{2},\] meaning that the moment of inertia of the solid cylinder is half the moment of inertia of the hollow cylinder.
03

Choose the correct answer

Since the moment of inertia of the hollow cylinder is larger than that of the solid cylinder, the correct answer is (c) The hollow cylinder will have the larger moment of inertia because its mass is located away from the axis of rotation.

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Most popular questions from this chapter

An object made of two disk-shaped sections, \(\mathrm{A}\) and \(\mathrm{B}\), as shown in the figure, is rotating about an axis through the center of disk A. The masses and the radii of disks \(A\) and \(B\). respectively are, \(2.00 \mathrm{~kg}\) and \(0.200 \mathrm{~kg}\) and \(25.0 \mathrm{~cm}\) and \(2.50 \mathrm{~cm}\). a) Calculate the moment of inertia of the object. b) If the axial torque due to friction is \(0.200 \mathrm{~N} \mathrm{~m}\), how long will it take for the object to come to a stop if it is rotating with an initial angular velocity of \(-2 \pi \mathrm{rad} / \mathrm{s} ?\)

A circular platform of radius \(R_{p}=4.00 \mathrm{~m}\) and mass \(M_{\mathrm{p}}=400 .\) kg rotates on frictionless air bearings about its vertical axis at 6.00 rpm. An 80.0 -kg man standing at the very center of the platform starts walking \((\) at \(t=0)\) radially outward at a speed of \(0.500 \mathrm{~m} / \mathrm{s}\) with respect to the platform. Approximating the man by a vertical cylinder of radius \(R_{\mathrm{m}}=0.200 \mathrm{~m}\) determine an equation (specific expression) for the angular velocity of the platform as a function of time. What is the angular velocity when the man reaches the edge of the platform?

A solid ball and a hollow ball, each with a mass of \(1.00 \mathrm{~kg}\) and radius of \(0.100 \mathrm{~m}\), start from rest and roll down a ramp of length \(3.00 \mathrm{~m}\) at an incline of \(35.0^{\circ} .\) An ice cube of the same mass slides without friction down the same ramp. a) Which ball will reach the bottom first? Explain! b) Does the ice cube travel faster or slower than the solid ball at the base of the incline? Explain your reasoning. c) What is the speed of the solid ball at the bottom of the incline?

A disk of clay is rotating with angular velocity \(\omega .\) A blob of clay is stuck to the outer rim of the disk, and it has a mass \(\frac{1}{10}\) of that of the disk. If the blob detaches and flies off tangent to the outer rim of the disk, what is the angular velocity of the disk after the blob separates? a) \(\frac{5}{6} \omega\) b) \(\frac{10}{11} \omega\) c) \(\omega\) d) \(\frac{11}{10} \omega\) e) \(\frac{6}{5} \omega\)

A thin uniform rod (length \(=1.00 \mathrm{~m},\) mass \(=2.00 \mathrm{~kg})\) is pivoted about a horizontal frictionless pin through one of its ends. The moment of inertia of the rod through this axis is \(\frac{1}{3} m L^{2} .\) The rod is released when it is \(60.0^{\circ}\) below the horizontal. What is the angular acceleration of the rod at the instant it is released?
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