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Ch 09: Rotation of Rigid Bodies
Young & Freedman Calc - University Physics 15th Edition
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 15th EditionCh 09: Rotation of Rigid BodiesProblem 34a
Chapter 9, Problem 34a

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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Step 1: Understand the concept of moment of inertia. The moment of inertia quantifies an object's resistance to rotational motion about a given axis. For a thin, uniform rod rotating about an axis perpendicular to its length and passing through its center, the formula is I = (1/12) * M * L^2, where M is the mass of the rod and L is its length.
Step 2: Convert the length of the rod from centimeters to meters, as SI units are required for calculations. Since 1 cm = 0.01 m, the length L = 60.0 cm = 0.60 m.
Step 3: Identify the given values: the mass of the rod M = 0.400 kg and the length L = 0.60 m. Substitute these values into the formula for the moment of inertia.
Step 4: Write the formula explicitly using MathML: I=112⁢M⁢L2. Substitute M = 0.400 kg and L = 0.60 m into the formula.
Step 5: Perform the calculation step-by-step (without solving for the final value): First, square the length L (L^2 = 0.60^2). Then multiply the squared length by the mass M. Finally, divide the result by 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, because the mass can be treated as evenly distributed along the rod. 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 an imaginary line around which an object rotates. For the given problem, the axis is located at the center of the rod and is 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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Related Practice
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 parallel to the bar through both 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 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

An airplane propeller is 2.08 m in length (from tip to tip) with mass 117 kg and is rotating at 2400 rpm (rev/min) about an axis through its center. You can model the propeller as a slender rod.

(a) What is its rotational kinetic energy?

(b) Suppose that, due to weight constraints, you had to reduce the propeller's mass to 75.0% of its original mass, but you still needed to keep the same size and kinetic energy. What would its angular speed have to be, in rpm?

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Textbook Question

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.)

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