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
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