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Ch 09: Rotation of Rigid Bodies
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 14th EditionCh 09: Rotation of Rigid BodiesProblem 29a
Chapter 9, Problem 29a

Calculate the moment of inertia of each of the following uniform objects about the axes indicated. Consult Table 9.2 as needed. A thin 2.50-kg rod of length 75.0 cm, about an axis perpendicular to it and passing through (i) one end and (ii) its center, and (iii) about an axis parallel to the rod and passing through it.

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Step 1: Recall the formula for the moment of inertia of a thin rod about different axes. For a thin rod of mass \( M \) and length \( L \): (i) About an axis perpendicular to the rod and passing through one end: \( I = \frac{1}{3} M L^2 \), (ii) About an axis perpendicular to the rod and passing through its center: \( I = \frac{1}{12} M L^2 \), and (iii) About an axis parallel to the rod and passing through it: \( I = 0 \) (since all mass is along the axis).
Step 2: Identify the given values in the problem. The mass of the rod is \( M = 2.50 \; \text{kg} \) and the length of the rod is \( L = 75.0 \; \text{cm} = 0.750 \; \text{m} \). Convert the length to meters for consistency in SI units.
Step 3: For part (i), substitute \( M = 2.50 \; \text{kg} \) and \( L = 0.750 \; \text{m} \) into the formula \( I = \frac{1}{3} M L^2 \). This will give the moment of inertia about an axis perpendicular to the rod and passing through one end.
Step 4: For part (ii), substitute \( M = 2.50 \; \text{kg} \) and \( L = 0.750 \; \text{m} \) into the formula \( I = \frac{1}{12} M L^2 \). This will give the moment of inertia about an axis perpendicular to the rod and passing through its center.
Step 5: For part (iii), note that the moment of inertia about an axis parallel to the rod and passing through it is \( I = 0 \), as all the mass is distributed along the axis itself. This completes the calculation for all parts of the problem.

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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 rotational motion about a specific axis. It depends on the mass distribution relative to that axis, with greater distances from the axis resulting in a higher moment of inertia. The formula for calculating it varies based on the shape and mass distribution of the object.
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Axis of Rotation

The axis of rotation is an imaginary line around which an object rotates. The choice of this axis significantly affects the moment of inertia, as it determines how the mass is distributed in relation to the axis. Different axes can lead to different moment of inertia values for the same object.
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Uniform Objects

Uniform objects have a consistent mass distribution throughout their volume or length. This property simplifies the calculation of the moment of inertia, as the mass can be treated as evenly spread. For example, a uniform rod has a constant density, making it easier to apply standard formulas for its moment of inertia.
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Related Practice
Textbook Question

An electric turntable 0.750 m in diameter is rotating about a fixed axis with an initial angular velocity of 0.250 rev/s and a constant angular acceleration of 0.900 rev/s2. What is the tangential speed of a point on the rim of the turntable at t = 0.200 s?

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Textbook Question

An electric turntable 0.750 m in diameter is rotating about a fixed axis with an initial angular velocity of 0.250 rev/s and a constant angular acceleration of 0.900 rev/s2. What is the magnitude of the resultant acceleration of a point on the rim at t = 0.200 s?

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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 that passes through the centers of the upper left and lower right spheres and through point O.

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Textbook 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 perpendicular to the bar through its center;

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Textbook 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 perpendicular to the bar through one of the balls;

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Textbook 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 parallel to the bar through both balls;

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