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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 31c
Chapter 9, Problem 31c

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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Identify the formula for the moment of inertia of the system about the given axis. Since the axis is parallel to the bar and passes through both balls, the moment of inertia will only depend on the two point masses (the balls). The bar itself does not contribute to the moment of inertia because it is parallel to the axis of rotation.
The moment of inertia for a point mass is given by the formula: I = mr2, where m is the mass of the point and r is the perpendicular distance from the axis of rotation.
In this case, each ball has a mass of 0.300 kg, and the distance of each ball from the axis of rotation is 0 m because the axis passes through both balls.
Substitute the values into the formula for each ball. For the first ball: I1 = 0.300 × 02. Similarly, for the second ball: I2 = 0.300 × 02.
Add the contributions of both balls to find the total moment of inertia: Itotal = I1 + I2. Since both terms are zero, the total moment of inertia is zero.

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Key Concepts

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

Moment of Inertia

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 the axis of rotation. For point masses, it is calculated as the product of the mass and the square of the distance from the axis of rotation. In this case, the moment of inertia will include contributions from both the bar and the balls at its ends.
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Parallel Axis Theorem

The parallel axis theorem allows us to calculate the moment of inertia of a body about any axis parallel to an 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 axis plus the product of the mass and the square of the distance between the two axes. This theorem is essential for determining the moment of inertia of the bar and the balls about the specified axis.
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Center of Mass

The center of mass is the point at which the mass of a system is concentrated and can be considered to act for translational motion. For composite systems, like the bar with balls, the center of mass can be found by considering the masses and their positions. Understanding the center of mass is crucial for applying the parallel axis theorem and calculating the moment of inertia accurately.
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Related Practice
Textbook Question

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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Textbook Question

You are a project manager for a manufacturing company. One of the machine parts on the assembly line is a thin, uniform rod that is 60.0 cm long and has mass 0.400 kg. What is the moment of inertia of this rod for an axis at its center, perpendicular to the rod?

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Textbook Question

You are a project manager for a manufacturing company. One of the machine parts on the assembly line is a thin, uniform rod that is 60.0 cm long and has mass 0.400 kg. One of your engineers has proposed to reduce the moment of inertia by bending the rod at its center into a V-shape, with a 60.0o angle at its vertex. What would be the moment of inertia of this bent rod about an axis perpendicular to the plane of the V at its vertex?

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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 and 0.500 m from it.

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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