Explanation This sphere consist both kinetic and potential energy. Use the law of conservation of energy. The gain in kinetic energy of the sphere is equal to the loss in potential energy. Case 1: Write the expression for conservation of energy for this system as. U i = K r o t + K t r a n s (I) Here, U i is the initial potential energy, K r o t is final rotational kinetic energy and K t r a n s is final translational kinetic energy. Write the expression for initial potential energy for the sphere as. U i = m g h Here, m is the mass of sphere, g is the acceleration under gravity and h is the height of the fallen sphere. Write the expression for final rotational kinetic energy as. K r o t = 1 2 I ω 2 (II) Here, I is the moment of inertia of sphere , K r o t and ω is the angular speed. The moment of inertia for the sphere is given by 2 5 m r 2 and the angular speed of the sphere is given by v r . Substitute 2 5 m r 2 for I and v r for ω in equation (II). K r o t = 1 2 ( 2 5 m r 2 ) ( v r ) 2 = 1 5 m v 2 Here, v is the linear velocity of sphere. Write the expression for final translational kinetic energy as. K t r a n s = 1 2 m v 2 Substitute m g h for U i , 1 5 m v 2 for K r o t and 1 2 m v 2 for K t r a n s in equation (I). m g h = 1 5 m v 2 + 1 2 m v 2 Simplify the above equation for velocity as. v = 10 7 g h (III) Case 2: In the case of no rolling or sphere sliding without friction, the rotation term is zero that means ω = 0

Physics for Scientists and Engineers with Modern, Revised Hybrid (with Enhanced WebAssign Printed Access Card for Physics, Multi-Term Courses)

9th Edition

ISBN: 9781305266292

Author: Raymond A. Serway, John W. Jewett

Publisher: Cengage Learning

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Chapter 10, Problem 60P

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The speed of the sphere and the time intervals required to reach the bottom in cases (a) and (b).

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Chapter 10, Problem 59P

Chapter 10, Problem 61P

Chapter 10 Solutions

Physics for Scientists and Engineers with Modern, Revised Hybrid (with Enhanced WebAssign Printed Access Card for Physics, Multi-Term Courses)

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The speed of the sphere and the time intervals required to reach the bottom in cases (a) and (b).

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Chapter 10 Solutions