13. Rotational Inertia & Energy

Parallel Axis Theorem

Open Question
A thin uniform rod of mass M and length L is bent at its center so that the two segments are now perpendicular to each other. Find its moment of inertia about an axis perpendicular to its plane and passing through (a) the point where the two segments meet
Open Question
Find the moment of inertia of a hoop (a thin-walled, hollow ring) with mass M and radius R about an axis perpendicular to the hoop's plane at an edge.
Open Question
A thin, rectangular sheet of metal has mass M and sidesof length a and b. Use the parallel-axis theorem to calculate themoment of inertia of the sheet for an axis that is perpendicular to theplane of the sheet and that passes through one corner of the sheet.
Open Question
A uniform bar has two small balls glued to its ends. The bar is 2.00 m long and has mass 4.00 kg, while the balls each have mass 0.300 kg and can be treated as point masses. Find the moment of inertia of this combination about an axis (d) parallel to the bar and 0.500 m from it.
Open Question
(II) Determine the moment of inertia of a 19-kg door around its hinges along one side. The door is 2.5 m high and 1.0 m wide. Ignore the thickness of the door.
Open Question
The sphere of mass M and radius R in FIGURE P12.75 is rigidly attached to a thin rod of radius r that passes through the sphere at distance 1/2 R from the center. A string wrapped around the rod pulls with tension T. Find an expression for the sphere's angular acceleration. The rod's moment of inertia is negligible.
Open Question
Figure 10–72 illustrates an H₂O molecule. The O―H bond length is 0.096 nm and the H―O―H bonds make an angle of 104°. Calculate the moment of inertia of the H₂O molecule (assume the atoms are points) about an axis passing through the center of the oxygen atom
(b) in the plane of the molecule, bisecting the H―O―H bonds.
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Open Question
A crucial part of a piece of machinery starts as a flat uniform cylindrical disk of radius R₀ and mass M. It then has a circular hole of radius R₁ drilled into it (Fig. 10–80). The hole’s center is a distance h from the center of the disk. Find the moment of inertia of this disk (with off-center hole) when rotated about its center, C. [Hint: Consider a solid disk and “subtract” the hole; use the parallel-axis theorem.]
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