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Ch 09: Rotation of Rigid Bodies
Chapter 9, Problem 9

A wagon wheel is constructed as shown in Fig. E9.33. The radius of the wheel is 0.300 m, and the rim has mass 1.40 kg. Each of the eight spokes that lie along a diameter and are 0.300 m long has mass 0.280 kg. What is the moment of inertia of the wheel about an axis through its center and perpendicular to the plane of the wheel? (Use Table 9.2.) Diagram of a wagon wheel showing rim, spokes, and hub for moment of inertia calculation.

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1
Identify the components of the wheel: the rim and the eight spokes.
Calculate the moment of inertia of the rim using the formula for a thin hoop: I_rim = m_rim * r^2, where m_rim is the mass of the rim and r is the radius.
Calculate the moment of inertia of one spoke using the formula for a thin rod rotating about one end: I_spoke = (1/3) * m_spoke * L^2, where m_spoke is the mass of one spoke and L is the length of the spoke.
Since there are eight spokes, multiply the moment of inertia of one spoke by 8 to get the total moment of inertia for all spokes: I_spokes_total = 8 * I_spoke.
Add the moment of inertia of the rim and the total moment of inertia of the spokes to get the total moment of inertia of the wheel: I_total = I_rim + I_spokes_total.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion about a specific axis. It depends on the mass distribution relative to that axis. For composite objects, the total moment of inertia is the sum of the moments of inertia of individual components, calculated using their respective mass and distance from the axis of rotation.
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Parallel Axis Theorem

The parallel axis theorem allows for the calculation of the moment of inertia of an object about any axis, given its moment of inertia about a parallel axis through its center of mass. It states that the moment of inertia about the new axis is equal to the moment of inertia about the center of mass plus the product of the mass and the square of the distance between the two axes.
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Composite Bodies

Composite bodies consist of multiple shapes or components, each contributing to the overall moment of inertia. In the case of the wagon wheel, the rim and spokes are treated as separate components, and their individual moments of inertia are calculated and summed to find the total moment of inertia of the wheel. Understanding how to analyze each part is crucial for accurate calculations.
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Related Practice
Textbook Question
A wheel is turning about an axis through its center with constant angular acceleration. Starting from rest, at t = 0, the wheel turns through 8.20 revolutions in 12.0 s. At t = 12.0 s the kinetic energy of the wheel is 36.0 J. For an axis through its center, what is the moment of inertia of the wheel?
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Textbook Question
Four small spheres, each of which you can regard as a point of mass 0.200 kg, are arranged in a square 0.400 m on a side and connected by extremely light rods (Fig. E9.28). Find the moment of inertia of the system about an axis

(c) that passes through the centers of the upper left and lower right spheres and through point O.
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Textbook 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
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Textbook Question
Compact Disc. A compact disc (CD) stores music in a coded pattern of tiny pits 10^-7 m deep. The pits are arranged in a track that spirals outward toward the rim of the disc; the inner and outer radii of this spiral are 25.0 mm and 58.0 mm, respectively. As the disc spins inside a CD player, the track is scanned at a constant linear speed of 1.25 m/s. (a) What is the angular speed of the CD when the innermost part of the track is scanned? The outermost part of the track?
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Textbook Question
A wheel of diameter 40.0 cm starts from rest and rotates with a constant angular acceleration of 3.00 rad/s^2. Compute the radial acceleration of a point on the rim for the instant the wheel completes its second revolution from the relationship (b) a_rad = v^2/r
696
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Textbook Question
A wheel of diameter 40.0 cm starts from rest and rotates with a constant angular acceleration of 3.00 rad/s^2. Compute the radial acceleration of a point on the rim for the instant the wheel completes its second revolution from the relationship (a) a_rad = ω^2r and (b) a_rad = v^2/r
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