To determine Determine the angular acceleration of the hoop cylinder assembly and the acceleration of the cart. Explanation Show the free body diagram of hoop and cart as in Figure (1). Consider the moment of inertia of the hoop about its center. I = m r 2 2 Here, mass moment of inertia of hoop is I , mass of the hoop is m and radius of the hoop is r . Apply equilibrium of force in the vertical direction for horizontal direction. ∑ F x = 0 F = M a a = F M Here, forces in the horizontal direction is F x , mass of cart is M and acceleration of the cart is a . Apply equilibrium of the force in the vertical direction for hoop. ∑ F y = 0 N − m g = 0 N = m g Here, forces in the vertical direction is F y , normal reaction is N and acceleration due to gravity is g . Consider the equilibrium for the hoop in the horizontal direction. ∑ F x = 0 F = m a G H a G H = F m Here, acceleration of the hoop is a G H . Consider the equilibrium for the hoop in the vertical direction. ∑ F y = 0 N = m g Consider there is no slipping condition for hoop. a = − ( a G H − α r ) (I) Here, angular acceleration is α . Take the positive about the center of the hoop. Assume anticlockwise as negative and clockwise as positive. ∑ M G H = I α − F r + m g r G H = I α (II) Here, distance of the mass center of the hoop from the geometrical center is r G H . Express linear acceleration of the cart. a = F m (III) Conclusion: Substitute F M for a and F m for a G H in Equation (I). F M = − F m + α r F M + F m = α r F ( M + m ) M m = α r F = r α M m ( M + m ) Substitute r α M m ( M + m ) for F , 4 r 3 π for r G H and m r 2 2 for I in Equation (II)

8th Edition

ISBN: 9781118885840

Author: James L. Meriam, L. G. Kraige, J. N. Bolton

Publisher: WILEY

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Chapter 6.5, Problem 86P

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Determine the angular acceleration of the hoop cylinder assembly and the acceleration of the cart.

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Chapter 6.5, Problem 85P

Chapter 6.5, Problem 87P

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