Explanation Given information: The total weight is 12 lb , the diameter of the semicircular plate is 16 in , the time is 2 sec and the length of the shaft A B is 40 in . Write the expression of mass. m = W g ....... (I) Here, the weight is W and the acceleration due to gravity is g . Write the expression of total moment of inertia in x direction for the assembly. I x = 1 4 m r 2 ....... (II) Here, the radius of the semicircular plate is r . Write the expression for the moment of the inertia about x y ′ -axis for vertical plate. I x y ′ = − m r 2 3 π ...... (III) Write the expression for the moment of the inertia about x z ′ -axis for vertical plate. I x z ′ = 0 Write the expression for the moment of the inertia about x y ′ -axis for horizontal plate. I x y ′ = 0 ...... (IV) Write the expression for the moment of the inertia about x z ′ -axis for horizontal plate. I x z ′ = − m r 2 3 π For assembly I x y ′ = I x z ′ . Write the expression for the moment of the inertia about x y ′ -axis for assembly. I x y ′ = − 2 m r 2 3 π ...... (V) Write the expression for the angular momentum about point C . H C = m r 2 ω ( 1 4 i + 2 3 π j + 2 3 π k ) Write the expression for the rate of change of the angular momentum about point C for the plane x y ′ z ′ . Write the expression of angular acceleration. Ω = ω j Write the expression for the rate of change of the angular momentum about point C . Substitute m r 2 α ( 1 4 i + 2 3 π j + 2 3 π k ) for for H C and ω j for Ω in Equation (VI). Write the expression for the moment about point A . M A = 5 r i × ( B y j + B z k ) + M 0 i Write the expression of moment about C . M A = H ˙ C ....... (VIII) Substitute 5 r i × ( B y j + B z k ) + M 0 i for M A and 1 4 m r 2 α i + 2 3 π m r 2 ( α − ω 2 ) j + 2 3 π m r 2 ( α − ω 2 ) k for H ˙ C in Equation (XVI). 5 r i × ( B y j + B z k ) + M 0 i = 1 4 m r 2 α i + 2 3 π m r 2 ( α − ω 2 ) j + 2 3 π m r 2 ( α − ω 2 ) k ...... (IX) Compare the term of i on both sides of equation (IX). M 0 = 1 4 m r 2 α ...... (X) Compare the term of j on both sides of equation (IX). − 5 r B z j = 2 3 π m r 2 ( α − ω 2 ) j B z = 2 3 π m r 2 ( α − ω 2 ) − 5 r B z = − 2 15 π m r ( α − ω 2 ) ...... (XI) Compare the term of k on both sides of equation (IX). 5 r B y k = 2 3 π m r 2 ( α + ω 2 ) k B y = 2 3 π m r 2 ( α + ω 2 ) 5 r B y = 2 15 π m r ( α + ω 2 ) ...... (XII) Write the expression for the angular acceleration. α = 2 θ t 2 ...... (XIII) Here, the time period is t and the angle is θ . Write the expression for the angular velocity. ω = α t ...... (XIV) Write the expression for the dynamic reaction at point B . B = B y j + B z k ....... (XV) Write the expression for the dynamic reaction at equilibrium condition. A + B = 0 ...... (XVI) Calculation: Substitute 32.2 ft / s 2 for g and 12 lb for W in Equation (I)

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ISBN: 9781259977237

Author: BEER

Publisher: MCG

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Chapter 18.2, Problem 18.76P

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The dynamic reactions at A.

The dynamic reactions at B.

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Chapter 18.2, Problem 18.75P

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