To determine The resulting velocity of each hemisphere. Explanation Given information: The potential energy of the compressed spring ( V ) is 120 J. The initial velocity of the assembly is v 0 of magnitude is ( v 0 ) 8 m/s. The angle when the cord is severed is 30 ° . The mass of hemisphere A ( m A ) is 2.5 kg. The mass of hemisphere B ( m B ) is 1.5 kg. Calculation: Draw the free body diagram of the two hemispheres as in Figure (1). Use reference frame moving with the mass centre. Determine the velocity v ′ A using the conservation of momentum. − m A v ′ A + m B v ′ B = 0 − m A v ′ A = − m B v ′ B v ′ A = m B m A v ′ B (1) Determine the velocity v ′ B using the conservation of energy. V = 1 2 m A ( v ′ A ) 2 + 1 2 m B ( v ′ B ) 2 Substitute m B m A v ′ B for v ′ A . V = 1 2 m A ( m B m A v ′ B ) 2 + 1 2 m B ( v ′ B ) 2 V = 1 2 m B ( v ′ B ) 2 ( m B m A + 1 ) V = m B ( m A + m B ) 2 m A ( v ′ B ) 2 ( v ′ B ) = 2 m A V m B ( m A + m B ) (2) Substitute 2.5 kg for m A , 1.5 kg for m B , and 120 J for V in Equation (2). ( v ′ B ) = 2 × 2.5 × 120 1.5 ( 2.5 + 1.5 ) = 10 m/s ( 30 ° − IQuad ) Determine the velocity at point A . Substitute 1.5 kg for m B , 2.5 kg for m A , and 10 m/s for v ′ B in Equation (1). v ′ A = ( 1.5 2.5 ) 10 = 6 m/s ( 30 ° − IIQuad ) Determine the velocity of hemisphere A using the relation. v A = [ v 0 → ] + [ v ′ A → ] Substitute 8 m/s for v 0 and 6 m/s ( 30 ° − IIQuad ) for v ′ A . v A = [ 8 m/s → ] + [ 6 m/s ( 30 ° − IIQuad ) ] Determine the magnitude of velocity of hemisphere A . v A = ( v A ) x 2 + ( v A ) y 2 Substitute ( 8 − 6 cos 30 ° ) m/s for ( v A ) x and ( 6 sin 30 ° ) m/s for ( v A ) y

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 14.38P

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Vector Mechanics for Engineers: Statics and Dynamics

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