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

Young & Freedman Calc 14th Edition

Ch 09: Rotation of Rigid Bodies

Problem 39

Chapter 9, Problem 39

The flywheel of a gasoline engine is required to give up 500 J of kinetic energy while its angular velocity decreases from 650 rev/min to 520 rev/min. What moment of inertia is required?

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Convert the angular velocities from revolutions per minute (rev/min) to radians per second (rad/s). Use the conversion factor: $ 1 \text{ rev} = 2\pi \text{ rad} $ and $ 1 \text{ min} = 60 \text{ s} $. For example, $ \omega = \text{(angular velocity in rev/min)} \times \frac{2\pi}{60} $.

Write the expression for the rotational kinetic energy of the flywheel: $ KE = \frac{1}{2} I \omega^2 $, where $ KE $ is the kinetic energy, $ I $ is the moment of inertia, and $ \omega $ is the angular velocity.

Calculate the change in kinetic energy, $ \Delta KE $, using the given values: $ \Delta KE = KE_{\text{initial}} - KE_{\text{final}} = 500 \text{ J} $.

Substitute the expressions for $ KE_{\text{initial}} $ and $ KE_{\text{final}} $ into $ \Delta KE $: $ \Delta KE = \frac{1}{2} I \omega_{\text{initial}}^2 - \frac{1}{2} I \omega_{\text{final}}^2 $.

Solve for the moment of inertia $ I $ by isolating it in the equation: $ I = \frac{2 \Delta KE}{\omega_{\text{initial}}^2 - \omega_{\text{final}}^2} $. Substitute the known values for $ \Delta KE $, $ \omega_{\text{initial}} $, and $ \omega_{\text{final}} $ to find $ I $.

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Key Concepts

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

Kinetic Energy of Rotation

The kinetic energy (KE) of a rotating object is given by the formula KE = 1/2 I ω², where I is the moment of inertia and ω is the angular velocity in radians per second. This concept is crucial for understanding how energy is stored in a rotating system and how it changes with variations in angular velocity.

Moment of Inertia

Moment of inertia (I) is a measure of an object's resistance to changes in its rotational motion, depending on the mass distribution relative to the axis of rotation. It plays a key role in determining how much torque is needed to change the angular velocity of an object, making it essential for solving problems involving rotational dynamics.

Angular Velocity Conversion

Angular velocity is often expressed in revolutions per minute (rev/min) but must be converted to radians per second (rad/s) for calculations involving kinetic energy. The conversion factor is 2π rad per revolution, and understanding this conversion is necessary to accurately apply the kinetic energy formula in the context of the problem.

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Textbook Question

A compound disk of outside diameter 140.0 cm is made up of a uniform solid disk of radius 50.0 cm and area density 3.00 g/cm² surrounded by a concentric ring of inner radius 50.0 cm, outer radius 70.0 cm, and area density 2.00 g/cm². Find the moment of inertia of this object about an axis perpendicular to the plane of the object and passing through its center.

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Textbook Question

If we multiply all the design dimensions of an object by a scaling factor f, its volume and mass will be multiplied by f³. By what factor will its moment of inertia be multiplied?

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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

A uniform 2.00-m ladder of mass 9.00 kg is leaning against a vertical wall while making an angle of 53.0° with the floor. A worker pushes the ladder up against the wall until it is vertical. What is the increase in the gravitational potential energy of the ladder?

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Textbook Question

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?

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Textbook Question

If we multiply all the design dimensions of an object by a scaling factor f, its volume and mass will be multiplied by f³. If a 1/48 scale model has a rotational kinetic energy of 2.5 J, what will be the kinetic energy for the full-scale object of the same material rotating at the same angular velocity?

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