(a) Explanation Given information: The length of the strings ( l ) is 9 inch. The relative velocity ( v ′ ) is 2.6 ft/s. The distance ( a ) is 1 ft. The initial velocity is (1.3 ft/s) i . Calculation: Draw the free body diagram of the system as shown in Figure (1). Express the linear momentum before break. L 0 = m v 0 + m v 0 + m v 0 = ( 3 m ) v 0 Here, linear momentum before break is L 0 , mass of each sphere is m , and velocity of the ring is v 0 . Substitute ( 1.3 ft / s ) i for v 0 . L 0 = ( 3 m ) ( 1.3 ft / s ) i = m ( 3.9 ft / s ) i (1) Express the linear momentum after break. L = ( m v A ) j − ( m v B ) j + ( m v C ) i Here, linear momentum after break is L , mass of each sphere is m , velocity of sphere A is v A , velocity of sphere B is v B , and velocity of sphere C is v C . After the ring breaks, A and B travel in the y direction and C in the x direction. Sphere A travels in the positive y axis direction and sphere B in the negative y axis. L = ( m v A ) j − ( m v B ) j + ( m v C ) i = ( m v C ) i + ( m v A − m v B ) j (2) According to the principle of impulse momentum, the linear momentum of a system is conserved. The liner momentum before the ring breaks is equal to the linear momentum after the ring breaks. Equate Equation (1) and (2). m ( 3.9 ft / s ) i = ( m v C ) i + ( m v A − m v B ) j (3) From Equation (3), equate j vectors. m v A − m v B = 0 m v A = m v B v A = v B From Equation (3), equate i vectors. ( m v C ) = m ( 3.9 ft / s ) v C = 3.9 ft / s Hence, the velocity of sphere C is 3.9 ft / s _ . According to the conservation of angular momentum, angular momentum of the system is conserved. The value for clockwise direction is positive. Sphere A travels in the positive y axis direction and sphere B in the negative y axis. ( 3 m ) l v ′ = − m v A x A + m v B ( x A + 1.0 ) + m v C d Here, relative velocity of ring is v ′ , velocity of A after break is v A , velocity of B after break is v B , velocity of C after break is v C , distance of A form y axis is x A , distance of B form y axis is x B , length of the string is l, and distance of C form x axis is d . The values of v A and v B are equal. Rewrite the equation: ( 3 m ) l v ′ = − m v A x A + m v A ( x A + 1 ) + m v C d ( 3 m ) l v ′ = − m v A x A + m v A x A + m v A + m v C d ( 3 m ) l v ′ = m v A ( 1 (b) To determine The distance ( d ):

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ISBN: 9781259633133

Author: BEER

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Chapter 14.2, Problem 14.55P

(a)

To determine

The velocity of each sphere.

(b)

To determine

The distance (d):

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Chapter 14.2, Problem 14.54P

Chapter 14.2, Problem 14.56P

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