Explanation Given information: The mass of the disk is 2.5 kg , radius of the disk is 80 mm , the angular velocity of the shaft is 12 rad / s , final angular velocity of the disk is 50 rad / s and angular acceleration is − 15 rad / s 2 . Expression for the mass moment of inertia about the x-axis. I ¯ x = 1 2 m r 2 ...... (I) Here, mass of the disk is m and radius of the disk is r . Expression for the mass moment of inertia about the y-axis. I ¯ y = 1 4 m r 2 ...... (II) Expression for angular momentum about A . H A = I ¯ x ω x ⋅ i + I ¯ y ω y ⋅ j ...... (III) Here angular velocity of disk about x-axis is ω x and angular velocity of disk about y-axis is ω y . Substitute 1 2 m r 2 for I ¯ x , ω 1 for ω x , ω 2 for ω y and 1 4 m r 2 for I ¯ y in Equation(III) H A = 1 2 m r 2 ω 1 ⋅ i + 1 4 m r 2 ω 1 ⋅ j ...... (IV) Expression for acceleration of mass center using C as fixed origin. a = r A / C ω 2 ....... (V) Here, r A / C is the distance from center of disk to the fixed origin C . Expression for r A / C r A / C = ( 120 mm ) i − ( 60 mm ) j ...... (VI) Expression for the rate of change of angular momentum with respect to axis of fixed orientation Expression for the rate of change of angular momentum with respect to the rotating frame A x y z . Here is α 1 the rate of change of angular velocity in x-direction. Write the vector form of angular momentum along the x-y direction. Ω = ω 2 k . Substitute 1 2 m r 2 α 1 j for , ω 2 k for Ω and ( 1 2 m r 2 ω 1 ⋅ i + 1 4 m r 2 ω 1 ⋅ j ) for H A in Equation (V) Expression for r A / D r A / D = ( ( − 150 mm ) k + ( 120 mm ) i − ( 60 mm ) j ) ...... (VIII) Expression for the force due to acceleration. F = m a ...... (IX) Expression for the force in vector form F = F x i + F y j ...... (X) Here, F x is component of force in x-direction and F y is component of force in y-direction. Expression for the moment about D ∑ M D = [ − ( 300 mm ) k ] × [ E x i + E y j ] ...... (XI) Here, reaction at E in x-direction is E x and reaction at E in y-direction is E y . Expression for effective moment about D ∑ ( M D ) e f f = H ˙ A + r A / D × F ...... (XII) Expression for equilibrium of forces in x-direction ∑ F x = D x + E x ...... (XIII) Expression for equilibrium of effective forces in x-direction ∑ ( F x ) e f f = F x ...... (XIV) Expression for equilibrium of forces in y-direction ∑ F y = ( D y + E y ) ...... (XV) Expression for equilibrium of effective forces in y-direction ∑ ( F y ) e f f = F y ...... (XVI) Expression for the reaction at point D D = D x i + D y j ...... (XVII) Expression for the reaction at point E E = E x i + E y j ...... (XVIII) Calculation: Substitute 2.5 kg for m and 80 mm for r , 50 rad / s for ω 1 , 12 rad / s for ω 2 and − 15 rad / s 2 for α 1 in Equation (VII) H ˙ A = 1 2 × 2.5 kg × ( 80 mm ) 2 ( − ( 50 rad / s ) ( 12 rad / s ) i + ( − 15 rad / s 2 ) j ) = 1 2 2.5 kg × ( 80 mm ( 1 m 1000 mm ) ) 2 ( − ( 50 rad / s ) ( 12 rad / s ) i + ( − 15 rad / s 2 ) j ) = ( − 4.8 N ⋅ m ) i − ( 0.120 N ⋅ m ) j Substitute ( − ( 4.8 N ⋅ m ) i + − ( 0.12 N ⋅ m ) j ) for H ˙ A in Equation (X) ∑ ( M D ) e f f = − ( 4.8 N ⋅ m ) i + − ( 0.12 N ⋅ m ) j + r A / D × F ...... (XII) Substitute ( 120 mm ) i − ( 60 mm ) j for r A / C and 12 rad / s for ω 2 in Equation (V) a = [ ( 120 mm × ( 1 m 1000 mm ) ) i − ( 60 mm × ( 1 m 1000 mm ) ) j ] ( 12 rad / s ) 2 = − [ ( 0.12 m ) i − ( 0.06 m ) j ] ( 12 rad / s ) 2 = − [ ( 17.28 m / s 2 ) i − ( 8

12th Edition

ISBN: 9781259977305

Author: BEER, Ferdinand P. (ferdinand Pierre), Johnston, E. Russell (elwood Russell), Cornwell, Phillip J., SELF, Brian P.

Publisher: Mcgraw-hill Education,

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Chapter 18.2, Problem 18.103P

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The dynamic reaction at D.

The dynamic reaction at E.

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Chapter 18.2, Problem 18.102P

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The dynamic reaction at D. The dynamic reaction at E.

Solution Summary: The author analyzes the dynamic reaction at D and E. The mass of the disk is 2.5kg, the angular velocity of a shaft, and the mass moment of inertia.