Physics for Scientists and Engineers: Foundations and Connections
Physics for Scientists and Engineers: Foundations and Connections
1st Edition
ISBN: 9781133939146
Author: Katz, Debora M.
Publisher: Cengage Learning
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Chapter 13, Problem 32PQ

(a)

To determine

The ratio of the rotational kinetic energy to the translational kinetic energy for each toy.

(a)

Expert Solution
Check Mark

Answer to Problem 32PQ

The ratio of the rotational kinetic energy to the translational kinetic energy for spherical toy is 25 and ratio of the rotational kinetic energy to the translational kinetic energy for cylindrical toy is 12

Explanation of Solution

Take R to represent the radius of both cylindrical toy as well as spherical toy, since radius will be cancelled in kinetic energy expression of each one. Given that translational speed is same for spherical and cylindrical toy.

Write the relation between linear and angular speed of rotation.

  v=Rω                                                                                                                    (I)

Here, R is the radius, v is the linear speed and ω is the angular speed.

Rearrange the equation for ω.

  ω=vR                                                                                                                     (II)

Write the expression for rotational kinetic energy of the spherical shaped toy

  Kr,sphere=12Isphereω2                                                                                                (III)

Here, Kr,sphere is the rotational kinetic energy of the sphere spherical shaped toy and I is the rotational inertia of the spherical shaped toy.

Write the expression for rotational inertia of the spherical shaped toy.

  Isphere=25MR2                                                                                                        (IV)

Here, M is the mass of the spherical shaped toy

Write the expression for rotational kinetic energy of the cylindrical shaped toy.

  Kr,cyl=12Icylω2                                                                                                       (V)

Here, Kr,cyl is the rotational kinetic energy of the small cylindrical shape toys and Icyl is the rotational inertia of the cylinder.

Write the expression for rotational inertia of the cylinder.

  Icyl=12MR2                                                                                                           (VI)

Here, M is the mass of the toys in cylindrical shaped toy

Write the expression for the translational kinetic energy of spherical shaped toy.

  Kt,sphere=12Mv2                                                                                                    (VII)

Write the expression for the translational kinetic energy of cylindrical shaped toy.

  Kt,cyl=12Mv2                                                                                                      (VIII)

Conclusion:

Substitute equations (II) and (IV) in equation (III).

  Kr,sphere=12(25MR2)(vR)2=12(25MR2)v2R2=210Mv2=15Mv2                                                                                    (IX)

Substitute equations (II) and (VI) in equation (V).

  Kr,cyl=12(12MR2)v2R2=14Mv2                                                                                            (X)

Divide equation (IX) by (VI) to get ratio of rotational kinetic energy to the translational kinetic energy for spherical shaped toy.

  Kr,sphereKt,sphere=15Mv212mv2=25

Divide equation (X) by (VII) to get ratio of rotational kinetic energy to the translational kinetic energy for cylindrical shaped toy.

  Kr,cylKt,cyl=14Mv212Mv2=12

Therefore, the ratio of the rotational kinetic energy to the translational kinetic energy for spherical toy is 25 and ratio of the rotational kinetic energy to the translational kinetic energy for cylindrical toy is 12 .

(b)

To determine

Comparison of the translational speeds of sphere and cylinder if they same angular speed instead of same translational speed.

(b)

Expert Solution
Check Mark

Answer to Problem 32PQ

The linear speed of cylinder is 0.65 times the linear speed of sphere.

Explanation of Solution

Write the expression for angular speed of sphere.

  ωsphere=vsphereRsphere                                                                                                       (XI)

Here, ωsphere is the angular speed of the sphere, vsphere is the linear speed of sphere and Rsphere is the radius of sphere.

Write the expression for angular speed of cylinder.

  ωcyl=vcylRcyl                                                                                                          (XII)

Here, ωcyl is the angular speed of cylinder, vcyl is the linear speed of cylinder and Rcyl is the radius of cylinder.

Conclusion:

In problem 31, it is given that radius of cylinder is 0.013m and radius of sphere is 0.020m . In problem, it is given that angular speed of sphere and cylinder are same.

Write condition given in question.

  ωsphere=ωcyl

Equate equation (XI) and (XII) to get vcylvsphere .

  vsphereRsphere=vcylRcylvcylvsphere=RcylRsphere

Substitute 0.013m for Rcyl and 0.020m for Rsphere in above equation to get vcylvsphere .

  vcylvsphere=0.013m0.020mvcyl=0.65vsphere

Therefore, the linear speed of cylinder is 0.65 times the linear speed of sphere.

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Chapter 13 Solutions

Physics for Scientists and Engineers: Foundations and Connections

Ch. 13 - Rotational Inertia Problems 5 and 6 are paired. 5....Ch. 13 - A 12.0-kg solid sphere of radius 1.50 m is being...Ch. 13 - A figure skater clasps her hands above her head as...Ch. 13 - A solid sphere of mass M and radius Ris rotating...Ch. 13 - Suppose a disk having massMtot and radius R is...Ch. 13 - Problems 11 and 12 are paired. A thin disk of...Ch. 13 - Given the disk and density in Problem 11, derive...Ch. 13 - A large stone disk is viewed from above and is...Ch. 13 - Prob. 14PQCh. 13 - A uniform disk of mass M = 3.00 kg and radius r =...Ch. 13 - Prob. 16PQCh. 13 - Prob. 17PQCh. 13 - The system shown in Figure P13.18 consisting of...Ch. 13 - A 10.0-kg disk of radius 2.0 m rotates from rest...Ch. 13 - Prob. 20PQCh. 13 - Prob. 21PQCh. 13 - In Problem 21, what fraction of the kinetic energy...Ch. 13 - Prob. 23PQCh. 13 - Prob. 24PQCh. 13 - Prob. 25PQCh. 13 - A student amuses herself byspinning her pen around...Ch. 13 - The motion of spinning a hula hoop around one's...Ch. 13 - Prob. 28PQCh. 13 - Prob. 29PQCh. 13 - Prob. 30PQCh. 13 - Sophia is playing with a set of wooden toys,...Ch. 13 - Prob. 32PQCh. 13 - A spring with spring constant 25 N/m is compressed...Ch. 13 - Prob. 34PQCh. 13 - Prob. 35PQCh. 13 - Prob. 36PQCh. 13 - Prob. 37PQCh. 13 - Prob. 38PQCh. 13 - A parent exerts a torque on a merry-go-round at a...Ch. 13 - Prob. 40PQCh. 13 - Today, waterwheels are not often used to grind...Ch. 13 - Prob. 42PQCh. 13 - A buzzard (m = 9.29 kg) is flying in circular...Ch. 13 - An object of mass M isthrown with a velocity v0 at...Ch. 13 - A thin rod of length 2.65 m and mass 13.7 kg is...Ch. 13 - A thin rod of length 2.65 m and mass 13.7 kg is...Ch. 13 - Prob. 47PQCh. 13 - Two particles of mass m1 = 2.00 kgand m2 = 5.00 kg...Ch. 13 - A turntable (disk) of radius r = 26.0 cm and...Ch. 13 - CHECK and THINK Our results give us a way to think...Ch. 13 - Prob. 51PQCh. 13 - Prob. 52PQCh. 13 - Two children (m = 30.0 kg each) stand opposite...Ch. 13 - A disk of mass m1 is rotating freely with constant...Ch. 13 - Prob. 55PQCh. 13 - Prob. 56PQCh. 13 - The angular momentum of a sphere is given by...Ch. 13 - Prob. 58PQCh. 13 - Prob. 59PQCh. 13 - Prob. 60PQCh. 13 - Prob. 61PQCh. 13 - Prob. 62PQCh. 13 - A uniform cylinder of radius r = 10.0 cm and mass...Ch. 13 - Prob. 64PQCh. 13 - A thin, spherical shell of mass m and radius R...Ch. 13 - To give a pet hamster exercise, some people put...Ch. 13 - Prob. 67PQCh. 13 - Prob. 68PQCh. 13 - The velocity of a particle of mass m = 2.00 kg is...Ch. 13 - A ball of mass M = 5.00 kg and radius r = 5.00 cm...Ch. 13 - A long, thin rod of mass m = 5.00 kg and length =...Ch. 13 - A solid sphere and a hollow cylinder of the same...Ch. 13 - A uniform disk of mass m = 10.0 kg and radius r =...Ch. 13 - When a person jumps off a diving platform, she...Ch. 13 - One end of a massless rigid rod of length is...Ch. 13 - A uniform solid sphere of mass m and radius r is...Ch. 13 - Prob. 77PQCh. 13 - A cam of mass M is in the shape of a circular disk...Ch. 13 - Prob. 79PQCh. 13 - Consider the downhill race in Example 13.9 (page...Ch. 13 - Prob. 81PQ
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