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Ch 09: Rotation of Rigid Bodies
All textbooksYoung & Freedman Calc 14th EditionCh 09: Rotation of Rigid BodiesProblem 9
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Chapter 9, Problem 9

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. (a) What is the moment of inertia of this rod for an axis at its center, perpendicular to the rod?

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Identify the formula for the moment of inertia (I) of a thin, uniform rod about an axis through its center, perpendicular to its length. The formula is: I = \frac{1}{12} M L^2, where M is the mass of the rod and L is the length of the rod.
Convert the length of the rod from centimeters to meters to ensure consistency in units, as the standard unit for length in physics is meters. Since 1 meter = 100 centimeters, convert 60.0 cm to meters.
Substitute the mass of the rod (0.400 kg) and its length in meters into the formula.
Calculate the value inside the formula by squaring the length of the rod and then multiplying by the mass.
Finally, multiply the result by the fraction \frac{1}{12} to find the moment of inertia.

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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 rotational motion about a specific axis. It depends on the mass distribution of the object relative to that axis. For a uniform rod, the moment of inertia can be calculated using the formula I = (1/12) * m * L^2, where m is the mass and L is the length of the rod.
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Intro to Moment of Inertia

Uniform Rod

A uniform rod is an object with a constant mass per unit length throughout its entire length. This uniformity simplifies calculations of physical properties, such as moment of inertia, since the mass can be treated as evenly distributed. In this case, the rod's length and mass are essential for determining its moment of inertia.
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Gravitational Force of Rod Parallel to Axis

Axis of Rotation

The axis of rotation is the line about which an object rotates. For the given problem, the axis is at the center of the rod and perpendicular to its length. The choice of axis significantly affects the moment of inertia, as different axes can lead to different values due to varying mass distributions relative to the axis.
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