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

Young & Freedman Calc 15th Edition

Ch 09: Rotation of Rigid Bodies

Problem 40

Chapter 9, Problem 40

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?

Verified step by step guidance

Step 1: Convert the number of revolutions into radians. Since one revolution corresponds to $2\pi$ radians, multiply the given number of revolutions (8.20) by $2\pi$ to find the total angular displacement $\theta$ in radians.

Step 2: Use the kinematic equation for rotational motion $\theta = \omega_0 t + \frac{1}{2} \alpha t^2$, where $\omega_0$ is the initial angular velocity (0 rad/s, since the wheel starts from rest), $\alpha$ is the angular acceleration, and $t$ is the time. Substitute $\theta$ and $t$ to solve for $\alpha$.

Step 3: Calculate the angular velocity $\omega$ at $t = 12.0\,\text{s}$ using the equation $\omega = \omega_0 + \alpha t$. Substitute $\omega_0 = 0$, $\alpha$ (from Step 2), and $t = 12.0\,\text{s}$ to find $\omega$.

Step 4: Use the rotational kinetic energy formula $K = \frac{1}{2} I \omega^2$, where $K$ is the kinetic energy (36.0 J), $I$ is the moment of inertia, and $\omega$ is the angular velocity (from Step 3). Rearrange the formula to solve for $I$.

Step 5: Substitute the values of $K$ and $\omega$ into the equation $I = \frac{2K}{\omega^2}$ to calculate the moment of inertia $I$.

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Key Concepts

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

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time. It is a vector quantity that indicates how quickly an object is rotating faster or slower. In this problem, the wheel experiences constant angular acceleration, which means its angular velocity increases uniformly from rest. This concept is crucial for determining the final angular velocity after a given time period.

Kinetic Energy of Rotation

The kinetic energy of a rotating object is given by the formula KE = 0.5 * I * ω², where I is the moment of inertia and ω is the angular velocity. This relationship shows how the energy of a rotating body depends on both its mass distribution (moment of inertia) and its rotational speed. In this question, knowing the kinetic energy at a specific time allows us to relate it to the moment of inertia once we find the angular velocity.

Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its rotational motion, analogous to mass in linear motion. It depends on the mass distribution relative to the axis of rotation. For the wheel in this problem, calculating the moment of inertia is essential to relate the kinetic energy and angular velocity, allowing us to solve for the unknown quantity using the provided data.

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Intro to Moment of Inertia

Related Practice

Textbook Question

A thin, rectangular sheet of metal has mass M and sides of length a and b. Use the parallel-axis theorem to calculate the moment of inertia of the sheet for an axis that is perpendicular to the plane of the sheet and that passes through one corner of the sheet.

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Textbook Question

A uniform sphere with mass $28.0$ kg and radius $0.380$ m is rotating at constant angular velocity about a stationary axis that lies along a diameter of the sphere. If the kinetic energy of the sphere is $236$ J, what is the tangential velocity of a point on the rim of the sphere?

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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 60.0^o 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

Find the moment of inertia of a hoop (a thin-walled, hollow ring) with mass M and radius R about an axis perpendicular to the hoop's plane at an edge.

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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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