Concept explainers
(a)
The ratio of the rotational kinetic energy to the translational kinetic energy for each toy.
(a)
Answer to Problem 32PQ
The ratio of the rotational kinetic energy to the translational kinetic energy for spherical toy is
Explanation of Solution
Take
Write the relation between linear and angular speed of rotation.
Here,
Rearrange the equation for
Write the expression for rotational kinetic energy of the spherical shaped toy
Here,
Write the expression for rotational inertia of the spherical shaped toy.
Here,
Write the expression for rotational kinetic energy of the cylindrical shaped toy.
Here,
Write the expression for rotational inertia of the cylinder.
Here,
Write the expression for the translational kinetic energy of spherical shaped toy.
Write the expression for the translational kinetic energy of cylindrical shaped toy.
Conclusion:
Substitute equations (II) and (IV) in equation (III).
Substitute equations (II) and (VI) in equation (V).
Divide equation (IX) by (VI) to get ratio of rotational kinetic energy to the translational kinetic energy for spherical shaped toy.
Divide equation (X) by (VII) to get ratio of rotational kinetic energy to the translational kinetic energy for cylindrical shaped toy.
Therefore, the ratio of the rotational kinetic energy to the translational kinetic energy for spherical toy is
(b)
Comparison of the translational speeds of sphere and cylinder if they same angular speed instead of same translational speed.
(b)
Answer to Problem 32PQ
The linear speed of cylinder is
Explanation of Solution
Write the expression for angular speed of sphere.
Here,
Write the expression for angular speed of cylinder.
Here,
Conclusion:
In problem 31, it is given that radius of cylinder is
Write condition given in question.
Equate equation (XI) and (XII) to get
Substitute
Therefore, the linear speed of cylinder is
Want to see more full solutions like this?
Chapter 13 Solutions
Student Solutions Manual For Katz's Physics For Scientists And Engineers: Foundations And Connections, Volume 1
- A thin stick of mass 0.2 kg and length L=0.5m is attached to the rim of a metal disk of mass M=2.0kg and radius R=0.3m . The stick is free to rotate around a horizontal axis through its other end (see the following figure). (a) If the combinatin is related with the stick horizontal, what is the speed of the center of the disk when the stick is vertical? (b) What is the acceleration fo the center of the disk at the stick is released? (c) At the instant the stick passes through the vertical?arrow_forwardThe angular position of a rod varies as 20.0t2radians from time t=0 . The rod has two beads on it as shown in the following figure, one at 10 cm from the ratation axis and the other at 20 cm from the rotation axis. (a) What is the instantaneous angular velocity of the rod at t=5s ? (b) What is the angular acceleration of the rod? (c) What are the tangential speeds of the beads at t=5s ? (d) What are the tangential acceleration of the beads at t=5s ? (e) What are the centripetal accelerations of the beads at t=5s ?arrow_forwardConsider again the two wind turbines in Problem 32. a. At what point along the 40-m blade would the angular speed of thepoint be the same as the angular speed at the end of the 20-mblade? Explain your answer. b. At what point along the 40-mblade would the translational speed of the point be the same asthe speed at the end of the 20-m blade? Explain your answer.arrow_forward
- A pulsar is a rapidly rotating neutron star. The Crab nebula pulsar in the constellation Taurus has a period of 33.510-3s , radius 10.0 km, and mass 2.81030kg . The pulsar’s rotational period will increase over time due to the release of electromagnetic radiation, which doesn’t change its radius but reduces its rotational energy. (a) What is the angular momentum of the pulsar? (b) Suppose the angular velocity decreases at a rate of 1014rad/s2 . What is the torque on the pulsar?arrow_forwardA system of point particles is shown in the following figure. Each particle has mass 0.3 kg and they all lie in the same place. (a) What is the moment of inertia of the system about the given axis? (b) If the system rotates at 5 rev/s, what is its rotational kinetic energy?arrow_forward(a) Calculate the rotational kinetic energy of Earth on its axis. (b) What is the rotational kinetic energy of Earth in its orbit around the Sun?arrow_forward
- While punting a football, a kicker rotates his leg about the hip joint. The moment of inertia of the leg is 3.75kgm2 and its rotational kinetic energy is 175 J. (a) What is the angular velocity of the leg? (b) What is the velocity of tip of the punter’s shoe if it is 1.05 m from the hip joint?arrow_forwardA vertical wheel with a diameter of 50 cm starts from rest and rotates with a constant angular acceleration of 5.0rad/s2 around a fixed axis through its center counterclockwise. (a) Where is the point that is initially at the bottom of the wheel at t=10s ? (b) What is the point’s linear acceleration at this instant?arrow_forwardDuring a 6.0-s time interval, a fly-wheel with a constant angular acceleration turns through 500 radians that acquire an angular velocity of 100 rad/s. (a) What is the angular velocity at the beginning of the 6.0 s? (b) What is the angular acceleration of the fly-wheel?arrow_forward
- A boy stands at the center of a platform that is rotating without friction at 1.0 rev/s. The boy holds weights as far from his body as possible. At this position the total moment of inertia of the boy, platform, and weights is 5.0kgm2 . The boy draws the weights in close to his body, thereby decreasing the total moment of inertia to 1.5kgm2 . (a) What Is the final angular velocity of the platform? (b) By how much does the rotational kinetic energy increase?arrow_forwardThe axis of Earth makes a 23.5 angle with a direction perpendicular to the plane of Earth’s orbit. As shown below, this axis precesses, making one complete rotation in 25,780 y. (a) Calculate the change in angular momentum in half this time. (b) What is the average torque producing this change in angular momentum? (c) If this torque were created by a pair of forces acting at the most effective point on the equator, what would the magnitude of each force be?arrow_forwardA bug of mass 0.020 kg is at rest on the edge of a solid cylindrical disk (M=0.10kg,R=0.10m) rotating in a horizontal plane around the vertical axis through its center. The disk is rotating at 10.0 rad/s. The bug crawls to the center of the disk. (a) What is the new angular velocity of the disk? (b) What is the change in the kinetic energy of the system? (c) If the bug crawls back to the outer edge of the disk, what is the angular velocity of the disk then? (d) What is the new kinetic energy of the system? (e) What is the cause of the increase and decrease of kinetic energy?arrow_forward
- Physics for Scientists and Engineers: Foundations...PhysicsISBN:9781133939146Author:Katz, Debora M.Publisher:Cengage LearningUniversity Physics Volume 1PhysicsISBN:9781938168277Author:William Moebs, Samuel J. Ling, Jeff SannyPublisher:OpenStax - Rice UniversityCollege PhysicsPhysicsISBN:9781938168000Author:Paul Peter Urone, Roger HinrichsPublisher:OpenStax College
- Physics for Scientists and Engineers, Technology ...PhysicsISBN:9781305116399Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningGlencoe Physics: Principles and Problems, Student...PhysicsISBN:9780078807213Author:Paul W. ZitzewitzPublisher:Glencoe/McGraw-HillPrinciples of Physics: A Calculus-Based TextPhysicsISBN:9781133104261Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning