Explanation Given information: The three sphere A , B , C each having mass is m . Calculation: Consider x is a distance between A and C . Calculate the horizontal distance ( x ) as shown below. l 2 = x 2 + ( l 3 ) 2 x 2 = l 2 − l 2 9 x 2 = 9 l 2 − l 2 9 x = 8 3 l Calculate angle θ as shown below. l sin θ = l 3 sin θ = 1 3 θ = 19.47 ° l cos θ = 8 3 l cos θ = 8 3 Sketch the velocity system as shown in Figure 1. Refer to Figure 1. Write the velocity vectors as follows: v 0 = − v 0 j v A = v A ( cos θ i − sin θ j ) v B / A = u B ( − sin θ i − cos θ j ) Express the velocity at B as follows: ( v B ) 2 = ( v A ) 2 + ( v B / A ) 2 v B = v A 2 + v B / A 2 Express the velocity at C as follows: v C = v C j Apply the conservation of momentum as follows. m v 0 = m v A + m v B / A + m v C = 2 m v A + m v B / A + m v C (1) Divided by m in Equation (1). m v 0 m = 2 m v A + m v B / A + m v C m v 0 = 2 v A + v B / A + v C (2) Substitute − v 0 j for v 0 , v A ( cos θ i − sin θ j ) for v A , v C j for v C , and u B ( cos θ i − cos θ j ) for v B / A in Equation (2). − v 0 j = 2 v A ( cos θ i − sin θ j ) + u B ( − sin θ i − cos θ j ) + v C j = v A cos θ i − v A sin θ j − u B sin θ i − u B cos θ j + v C j Resolving the components of i, j coordinates as follows: 2 v A cos θ − u B sin θ = 0 (3) v 0 = 2 v A sin θ + u B cos θ − v C (4) Solve Equation (3) and (4) as follows: v A = 1 6 ( v 0 + v C ) u B = 0.94281 ( v 0 + v C ) Find the velocity of sphere C using conservation of energy as follows: 1 2 m v 0 2 = 1 2 m v A 2 + 1 2 m v B 2 + 1 2 m v C 2 = 1 2 m v A 2 + 1 2 m ( v A 2 + u B 2 ) + 1 2 m v C 2 (5) Divided by 1 2 m in Equation (5)

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 14.2, Problem 14.43P

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Find the velocity of each sphere after impact.

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Chapter 14.2, Problem 14.42P

Chapter 14.2, Problem 14.44P

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Chapter 14 Solutions

Vector Mechanics for Engineers: Statics and Dynamics

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Find the velocity of each sphere after impact.

Solution Summary: The author calculates the velocity of spheres A, B, and C after impact.

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Chapter 14 Solutions