(a) Explanation Given information: The diameter of the ball is d = 8 in . The weight of the ball is W = 12 lb . The forward velocity is v 0 = 15 ft / s . The backspin velocity is ω 0 = 18 rad / s . The coefficient of kinetic friction between the ball and the alley is μ k = 0.10 . Assumption: Apply the sign convention for calculating the equations of equilibrium as below. For the horizontal forces equilibrium condition, take the force acting towards right side as positive ( → + ) and the force acting towards left side as negative ( ← − ) . For the vertical forces equilibrium condition, take the upward force as positive ( ↑ + ) and downward force as negative ( ↓ − ) . For moment equilibrium condition, take the clockwise moment as negative and counter clockwise moment as positive. Calculation: Consider the acceleration due to gravity as g = 32.2 ft / s 2 . The radius of gyration of the solid ball is k ¯ = 2 5 r . Calculate the moment of inertia ( I ¯ ) as shown below. I ¯ = m k ¯ 2 Substitute 2 5 r for k ¯ . I ¯ = m × ( 2 5 r ) 2 = 2 5 m r 2 Sketch the Free Body Diagram of the ball as shown in Figure 1. Refer to Figure 1. Apply the Equations of Equilibrium as shown below. Apply the Equilibrium of forces along x direction as shown below. ∑ F x = m a x − μ k m g = m a ¯ a ¯ = μ k g ← (1) Apply the Equilibrium of moment about G as shown below. ∑ M G = I G α + m a d μ k m g r = I ¯ α Substitute 2 5 m r 2 for I ¯ . μ k m g r = 2 5 m r 2 α μ k g r = 2 5 r 2 α α = 5 μ k g 2 r Calculate the velocity ( v ) as shown below. v = r ω Calculate the time ( t 1 ) using the relation of velocity as shown below. v = v 0 − a ¯ t Substitute μ k g for a ¯ . v ¯ = v 0 − μ k g t (b) To determine The speed of the ball at time t 1 . (c) To determine The distance the ball will have travelled at time t 1 .

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Chapter 16.1, Problem 16.72P

(a)

To determine

The time at which the ball will start rolling without sliding.

(b)

To determine

The speed of the ball at time t1.

(c)

To determine

The distance the ball will have travelled at time t1.

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Chapter 16.1, Problem 16.71P

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The time at which the ball will start rolling without sliding.

Solution Summary: The author calculates the coefficient of kinetic friction between the ball and the alley using the sign convention.

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The time at which the ball will start rolling without sliding.

The speed of the ball at time t1.

The distance the ball will have travelled at time t1.

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Chapter 16 Solutions