(a) Explanation Given information: The mass of the cylinder A is m = 4 kg . The radius of the cylinder A is r = 150 mm . The angular velocity in the cylinder A is ω 0 = 50 rad/s . The number of revolution executed to attain constant velocity by cylinder A is θ A = 3 rev . The number of revolution executed to attain constant velocity by cylinder B is θ B = 1 rev . Calculation: Consider the acceleration due to gravity is g = 9.81 m/s 2 . Find the mass moment of inertia ( I ¯ ) of the cylinder using the equation. I ¯ = 1 2 m r 2 Substitute 4 kg for m and 150 mm for r . I ¯ = 1 2 × 4 × ( 150 mm × 1 m 1,000 mm ) 2 = 0.045 kg ⋅ m 2 Cylinder A : Show the free-body diagram of the cylinder A as in Figure 1. Resolve the vertical component of forces to find the normal forces. ∑ F y = 0 N − W = 0 N = m g Substitute 4 kg for m and 9.81 m/s 2 for g . N = 4 × 9.81 = 39.24 N Find the frictional force F using the relation. F = μ k N Here, the coefficient of kinetic friction is μ k . Substitute 39.24 N for N . F = 39.24 μ k Find the moment at cylinder A by taking moment about point A . ∑ M A = 0 M A = F r Substitute 39.24 μ k for F and 150 mm for r . M A = 39.24 μ k × 150 mm × 1 m 1,000 mm = 5.886 μ k N ⋅ m Initial stage (before contact with cylinder B ); Find the total kinetic energy ( T 1 ) using the relation. T 1 = 1 2 I ¯ ω 0 2 Substitute 0.045 kg ⋅ m 2 for I ¯ and 50 rad/s for ω 0 . T 1 = 1 2 × 0.045 × 50 2 = 56.25 N ⋅ m Final stage (After contact with cylinder B ): Find the final total kinetic energy ( T 2 ) using the relation. T 2 = 1 2 I ¯ ω A 2 Substitute 0.045 kg ⋅ m 2 for I ¯ . T 2 = 1 2 × 0.045 × ω A 2 = 0.0225 ω A 2 Find the work done ( U 1 → 2 ) for cylinder A to be in contact with cylinder B as follows; U 1 → 2 = − M A θ A Substitute 5.886 μ k N ⋅ m for M A and 3 rev for θ A . U 1 → 2 = − 5.886 μ k × 3 rev × 2 π rad 1 rev = − 110.9485 μ k N ⋅ m Write the equation of work and energy for the system using the equation. T 1 + U 1 → 2 = T 2 Substitute 56.25 N ⋅ m for T 1 , − 110.9485 μ k N ⋅ m for U 1 → 2 , and 0.0225 ω A 2 for T 2 56 (b) To determine Find the coefficient of kinetic friction μ k .

Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259638091

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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Chapter 17.1, Problem 17.8P

(a)

To determine

Find the final angular velocity of each cylinder.

(b)

To determine

Find the coefficient of kinetic friction μk.

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Find the final angular velocity of each cylinder.

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Find the coefficient of kinetic friction μk.

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