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Ch. 10 - Rotational Motion
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 10 - Rotational MotionProblem 70
Chapter 10, Problem 70

A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping down a lane at 3.7 m/s. Calculate its total kinetic energy.

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Identify the types of kinetic energy involved: Since the bowling ball is rolling without slipping, it has both translational kinetic energy and rotational kinetic energy.
Write the formula for total kinetic energy: \( KE_{\text{total}} = KE_{\text{translational}} + KE_{\text{rotational}} \). Translational kinetic energy is given by \( KE_{\text{translational}} = \frac{1}{2} m v^2 \), and rotational kinetic energy is given by \( KE_{\text{rotational}} = \frac{1}{2} I \omega^2 \), where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.
Determine the moment of inertia for a solid sphere: For a solid sphere rolling about its axis, the moment of inertia is \( I = \frac{2}{5} m r^2 \), where \( m \) is the mass and \( r \) is the radius of the sphere.
Relate angular velocity \( \omega \) to linear velocity \( v \): Since the ball rolls without slipping, \( \omega = \frac{v}{r} \). Substitute this expression for \( \omega \) into the rotational kinetic energy formula.
Substitute all known values into the total kinetic energy formula: Use \( m = 7.3 \ \text{kg} \), \( r = 0.09 \ \text{m} \), and \( v = 3.7 \ \text{m/s} \) to calculate \( KE_{\text{total}} = \frac{1}{2} m v^2 + \frac{1}{2} \left( \frac{2}{5} m r^2 \right) \left( \frac{v}{r} \right)^2 \). Simplify the expression to find the total kinetic energy.

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Key Concepts

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

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 1/2 mv², where m is the mass and v is the velocity. For rolling objects, both translational and rotational kinetic energy must be considered.
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Intro to Rotational Kinetic Energy

Rolling Motion

Rolling motion occurs when an object rotates about an axis while translating along a surface. For a solid sphere, the total kinetic energy is the sum of translational kinetic energy and rotational kinetic energy, which is given by KE_rotational = 1/2 Iω², where I is the moment of inertia and ω is the angular velocity.
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Conservation of Energy in Rolling Motion

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotation and depends on the mass distribution relative to the axis of rotation. For a solid sphere, the moment of inertia is I = 2/5 mR², where m is the mass and R is the radius, which is crucial for calculating the rotational kinetic energy in rolling objects.
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Related Practice
Textbook Question

Two blocks are connected by a light string passing over a pulley of radius 0.15 m and moment of inertia I. The blocks move (towards the right) with an acceleration of 1.00 m/s² along their frictionless inclines (see Fig. 10–62). Find the net torque acting on the pulley, and determine its moment of inertia, I.

Textbook Question

A solid rubber ball rests on the floor of a railroad car when the car begins moving with acceleration a. Assuming the ball rolls without slipping, what is its acceleration relative to the car?

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Textbook Question

(III) Integrate to derive the formula for the moment of inertia of a uniform thin rod of length ℓ about an axis through its center, perpendicular to the rod (see Fig. 10–21f).

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Textbook Question

A 2.30-m-long pole is balanced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]

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Textbook Question

A solid rubber ball rests on the floor of a railroad car when the car begins moving with acceleration a. Assuming the ball rolls without slipping, what is its acceleration relative to the ground?

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Textbook Question

II) A uniform solid sphere of radius r0 = 24.5 cm and mass m = 1.60 kg starts from rest and rolls without slipping down a 30.0° incline that is 10.0 m long. Calculate its translational and rotational speeds when it reaches the bottom. Avoid putting in numbers until the end so you can answer.

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