Explanation Given information: The mass ( m A ) of the sphere A is 2 kg. The radius ( r ) of the sphere A is 40 mm. The velocity ( v ¯ 1 ) of the sphere A before impact is 2 m/s. The mass ( m B ) of the bar B is 0.5 kg. The length ( L ) of the bar B is 100 mm. The coefficient of restitution ( e ) between A and B is 0.1. The coefficient of kinetic friction between the sphere and the horizontal surface is μ k . Calculation: Find the angular velocity ( ω 1 ) of the sphere A before the impact using the equation: ω 1 = v ¯ 1 r Substitute 2 m / s for v ¯ 1 , and 40 mm for r . ω 1 = 2 40 mm × 1 m 1 , 000 mm = 50 rad / s Consider the conservation of momentum principle. Syst Momenta 1 + Syst Ext Imp 1y2 = Syst Momenta 2 Sketch the impulse and momentum diagram of the system as shown in Figure (1). Here, ∫ P d t is the magnitude of impulse acting between the sphere A and slender bar B , d t is the time interval, ω A is the angular velocity of the sphere A immediately after the impact, I ¯ A is the centroidal moment of inertia of the sphere A immediately after the impact, v ¯ A is the velocity of the sphere A immediately after the impact, ω B is the angular velocity of the slender bar B immediately after the impact, I ¯ B is the centroidal moment of inertia of the slender bar B immediately after the impact, v ¯ B is the velocity of the slender bar B immediately after the impact, and b is the distance between points B and Q . Refer Figure (1), Take moment about center A considering only sphere. I ¯ A ω 1 + 0 = I ¯ A ω A I ¯ A ω 1 = I ¯ A ω A ω A = ω 1 Substitute 50 rad / s for ω 1 . ω A = 50 rad / s Thus, the angular velocities of the sphere immediately after the impact is 50.0 rad/s _ . Calculate the distance ( b ). b = L 2 − r Substitute 100 mm for L , and 40 mm for r . b = 100 2 − 40 = 10 mm × 1 m 1 , 000 mm = 0.010 m Write the centroidal moment of inertia ( I ¯ B ) of the slender bar B . I ¯ B = 1 12 m B L 2 Write the equation of velocity ( v Q ) at the contact point Q using kinematics: v Q = v ¯ B + b ω B Consider the impact condition for the given system. v ¯ A − v Q = − e v ¯ 1 Substitute v ¯ B + b ω B for v Q . v ¯ A − ( v ¯ B + b ω B ) = − e v ¯ 1 v ¯ A − v ¯ B − b ω B = − e v ¯ 1 Substitute 0.010 m for b , 0.1 for e , and 2 m / s for v ¯ 1

Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9780073398242

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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Chapter 17.3, Problem 17.129P

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Find the angular velocities of the sphere and the bar immediately after the impact.

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Chapter 17.3, Problem 17.128P

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Chapter 17 Solutions

Vector Mechanics for Engineers: Statics and Dynamics

Ch. 17.1 - A round object of mass m and radius r is released...Ch. 17.1 - Prob. 17.2CQ Ch. 17.1 - Prob. 17.3CQ Ch. 17.1 - Prob. 17.4CQ Ch. 17.1 - Slender bar A is rigidly connected to a massless...Ch. 17.1 - A 200-kg flywheel is at rest when a constant 300...Ch. 17.1 - The rotor of an electric motor has an angular...Ch. 17.1 - Prob. 17.3P Ch. 17.1 - Two disks of the same material are attached to a...Ch. 17.1 - Prob. 17.5P

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Find the angular velocities of the sphere and the bar immediately after the impact.

Solution Summary: The author calculates the angular velocities of sphere and bar immediately after the impact using the equation: c

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Find the angular velocities of the sphere and the bar immediately after the impact.

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