To determine The coefficient of restitution ( e ) between the two spheres. Explanation Given information: The weight of the sphere A ( W A ) is 50 lb. The weight of the sphere B ( W B ) is 4.6 lb. The radius of sphere A ( r A ) is 4.5 in. The radius of sphere B ( r B ) is 2.0 in. The initial velocity of the sphere A ( v A ) is 6 ft/s. The maximum height of the sphere B swing ( h ) is 0.75 ft. Calculation: Calculate the mass of the sphere A ( m A ) using the formula: W A = m A g Here, g is the acceleration due to gravity. Substitute 50 lb for W A and 32.2 ft/s 2 for g . 50 = m A ( 32.2 ) m A = 50 32.2 = 1.5528 lb ⋅ s 2 /ft Calculate the mass of the sphere B ( m B ) using the formula: W B = m B g Substitute 4.6 lb for W A and 32.2 ft/s 2 for g . 4.6 = m B ( 32.2 ) m B = 4.6 32.2 m B = 0.14286 lb ⋅ s 2 /ft Show the sphere A and sphere B with the dimension as Figure (1). Calculate the horizontal distance from B to C using the relation: C B = ( r A + r B ) 2 − ( r A − r B ) 2 Substitute 4.5 in. for r A and 2.0 in. for r B . C B = ( 4.5 + 2 ) 2 − ( 4.5 − 2 ) 2 = 6 in . Show the diagram of the angle of impulse force from geometry of A and B as in Figure (2). Calculate the angle of impulse force ( θ ) from geometry of A and B using the relation: tan θ = r B C B Substitute 2.0 in. for r B and 6 in. for CB . θ = tan − 1 ( 2.5 6 ) = 22.62 ° Show the impulse momentum diagram for the spheres A as Figure (3). Write the expression for the impulse momentum equation for sphere A as follows: m A v A sin θ + 0 = m A ( v ′ A ) x sin θ + m A ( v ′ A ) y cos θ v A sin θ + 0 = ( v ′ A ) x sin θ + ( v ′ A ) y cos θ Divide the above equation with cos θ . v A sin θ cos θ = ( v ′ A ) x sin θ cos θ + ( v ′ A ) y cos θ cos θ v A tan θ = ( v ′ A ) x tan θ + ( v ′ A ) y Substitute 22.62 ° for θ and 6 ft/s for v A . 6 × tan 22.62 ° = ( v ′ A ) x tan 22.62 ° + ( v ′ A ) y 2.5 = 0.41667 ( v ′ A ) x + ( v ′ A ) y 2.5 ( v ′ A ) x + 6 ( v ′ A ) y = 15 (1) Show the impulse momentum diagram for the spheres B as Figure (4)

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 13, Problem 13.194RP

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The coefficient of restitution (e) between the two spheres.

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Chapter 13, Problem 13.193RP

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Vector Mechanics for Engineers: Statics and Dynamics

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