Explanation Given information: The weight ( W ) of the assembly is 12 lb. The diameter ( d ) of the semicircular plate is 16 inch. The initial angular velocity ( ω 0 ) of the assembly is 0. The initial angle of rotation ( θ 0 ) of the assembly is 0. The length ( L ) of the shaft is 40 in. The half of the length ( b ) of the shaft is 20 in.. Calculation: Write the vector form of angular velocity. ω ˙ = ω i Write the vector form of angular acceleration. α = α i Consider fixed axis rotation with constant angular acceleration. Find the angular acceleration of the assembly when t = 2 s using the equation: θ = θ 0 + ω 0 t + 1 2 α t 2 Substitute 2 π for θ , 0 for θ 0 , 0 for ω 0 , and 2 s for t . 2 π = 0 + ( 0 ) ( 2 ) + 1 2 α ( 2 ) 2 2 π = 0 + 0 + 2 α α = 3.1416 rad/s 2 Find the angular velocity of the assembly at t = 2 s using kinematics. ω = ω 0 + α t Substitute 0 for ω 0 , 3.1416 rad/s 2 for α and 2 s for t . ω = 0 + ( 3.1416 ) ( 2 ) = 6.2832 rad/s Write the equation of angular momentum ( H G ) about origin at C . H C = I x ω i − I x y ω j − I x z ω k (1) Write the equation of angular momentum [ ( H ˙ C ) A x y z ] about the reference frame using Equation (1). H C = I ¯ x ω i − I ¯ x y ω j − I ¯ x z ω k Substitute ( H ˙ C ) A x y z for H C and α for ω . ( H ˙ C ) C x y z = I x α i − I x y α j − I x z α k Write the equation of angular velocity ( Ω ) about the reference frame. Ω = ω i Write the equation of rate of change of angular momentum ( H ˙ G ) of the system about C . H ˙ C = ( H ˙ C ) C x y z + Ω × H C Substitute I ¯ x α i − I ¯ x y α j − I ¯ x z α k for ( H ˙ G ) A x y z , ω i for Ω , and I ¯ x ω i − I ¯ x y ω j − I ¯ x z ω k for H C . H ˙ C = ( I ¯ x α i − I ¯ x y α j − I ¯ x z α k ) + ( ω i ) × ( I ¯ x ω i − I ¯ x y ω j − I ¯ x z ω k ) = I ¯ x α i − I ¯ x y α j − I ¯ x z α k + | i j k ω 0 0 I ¯ x ω − I ¯ x y ω − I ¯ x z ω | = I ¯ x α i − I ¯ x y α j − I ¯ x z α k + ( − ω ) [ ( − I ¯ x z ω j ) − ( − I ¯ x y ω k ) ] = I ¯ x α i − I ¯ x y α j − I ¯ x z α k + I ¯ x z ω 2 j − I ¯ x y ω 2 k (2) Sketch the free body diagram of the system as shown in Figure (1). Refer Figure (1). Find and tabulate the required moments and products of inertia as in Table (1). Part A I x I x y I x z 1 1 2 π r 2 1 8 π r 4 − 2 3 r 4 0 2 1 2 π r 2 1 8 π r 4 0 − 2 3 r 4 3 1 2 π r 2 1 8 π r 4 0 − 2 3 r 4 4 1 2 π r 2 1 8 π r 4 − 2 3 r 4 0 Σ 2 π r 2 1 2 π r 2 − 4 3 r 4 − 4 3 r 4 Find the mass of the system. m = W g Substitute 12 lb for W and 32.2 ft/s 2 for g . m = 12 32.2 = 0.37267 lb ⋅ s 2 /ft Find the radius of the semicircular plate using the equation: r = d 2 Substitute 16 in. for d . r = 16 in × 1 in . 12 ft 2 = 0.66667 ft Refer Table (1), Find the density ( ρ ) of the system using the equation: ρ = m A Substitute 2 π r 2 for A . ρ = m 2 π r 2 Substitute 0

Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9780073398242

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

Publisher: McGraw-Hill Education

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Chapter 18.2, Problem 18.76P

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The dynamic reactions at A and B at t=2 s.

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The dynamic reactions at A and B at t = 2 s .

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