Explanation Given information: The total weight is 2.7 lb , the angular acceleration is 150 rad / s 2 and the angular velocity is 240 rpm . 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 area of the sheet metal. A = 2 × A 1 + 2 × A 2 ....... (II) Here, the area of rectangle is A 1 and the area of triangle is A 2 . Write the expression of area of rectangle. A 1 = b × b Here, the length of the rectangle is is b and the width of the rectangle is is b . Write the expression of area of triangle. A 2 = 1 2 b × b Here, the base of the triangle is b and the height of the triangle is b . Substitute b × b for A 1 and 1 2 b × b for A 2 in Equation (II). A = 2 × b × b + 2 × ( 1 2 b × b ) = 2 b 2 + b 2 = 3 b 2 ...... (III) Write the expression of mass per unit area of the system. ρ = m A ....... (IV) The below figure represent the schematic diagram of the schematic diagram of the sheets Figure-(1) Write the expression of mass moment of inertia of the rectangular sheet in x direction. ( I x ) mass = ρ ( I x ) area ....... (V) Here, the moment of inertia in x direction is ( I x ) area . Write the expression of moment of inertia in x direction. ( I x ) area = 1 3 b 4 + 1 3 b 4 = 2 3 b 4 ....... (VI) Write the expression of mass moment of inertia of the rectangular sheet in x y direction. ( I x y ) mass = ρ ( I x y ) area ....... (VII) Write the expression of moment of inertia of the rectangular sheet in x y direction. ( I x y ) area = − 1 2 b 4 ....... (VIII) Here, the moment of inertia in x y direction is ( I x y ) area . The below figure represent area of triangular sheets. Figure-(2) Write the expression of mass moment of inertia of the triangular sheet in x direction. ( I x 1 ) mass = ρ ( I x 1 ) area ....... (IX) Here, the moment of inertia in x direction is ( I x 1 ) area . Write the expression of moment of inertia in x direction. ( I x 1 ) area = 1 12 b 4 + 1 12 b 4 = 1 6 b 4 ....... (X) Write the expression of mass moment of inertia of the triangle sheet in x z direction. ( I x z 1 ) mass = ρ ( I x z 1 ) area ....... (XI) Here, the moment of inertia in x z direction is ( I x z ) area . Write the expression of moment of inertia in x z direction. ( I x z 1 ) area = − ( 2 3 b 4 − 1 4 b 4 ) = − 5 12 b 4 ........ (XII) Write the expression of total moment of inertia in x direction. I x = ( I x ) mass + ( I x 1 ) mass ....... (XIII) Write the expression of total moment of inertia in x y direction. I x y = ( I x y ) mass ....... (XIV) Write the expression of total moment of inertia in x z direction. I x z = ( I x z 1 ) mass ....... (XV) Write the expression of rate of change of angular momentum. H ˙ A = ( H ˙ A ) A x y z + Ω × H A ........ (XVI) Here, the angular momentum of reference frame A x y z is ( H ˙ A ) A x y z , the angular velocity of the frame A x y z is Ω and the angular momentum about A is H A . Write the expression of angular momentum of reference frame A x y z . ( H A ) A x y z = I x α i − I x y α j − I x z α k Here, the angular acceleration is α , the unit vector along the x ′ axis is i ′ , the unit vector along the y ′ axis is j ′ and the unit vector along the z ′ axis is k ′ . Write the expression of angular momentum about A . H A = I x ω i − I x y ω j − I x z ω k Here, the angular velocity is ω . Write the expression of angular velocity of the frame A x y z . Ω = ω j Substitute ω j for Ω , I x ω i − I x y ω j − I x z ω k for H A and I x α i − I x y α j − I x z α k for ( H A ) A x y z in Equation (XV). H ˙ A = ( 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 ) + { ( ω i ) × ( I x ω i ) − ( ω i ) × ( I x y ω j ) − ( ω i ) × ( I x z ω k ) } = I x α i − I x y α j − I x z α k + − I x y ω 2 k + I x z ω 2 j = I x α i − ( I x y α − I x z ω 2 ) j − ( I x z α + I x y ω 2 ) k Write the expression of moment about A . M A = M 0 i + 4 b i × ( B y j + B z k ) = M 0 i + 4 b B y i × j + 4 b B z i × k = M 0 i + 4 b B y k − 4 b B z j Here, the y component of the reaction at B is B y and the z component of the reaction at B is B z . Write the expression of moment about A . M A = H ˙ A To determine (b) The dynamic reactions at A . The dynamic reactions at B .

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 18.73P

To determine

(a)

The couple M0.

To determine

(b)

The dynamic reactions at A.

The dynamic reactions at B.

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Chapter 18.2, Problem 18.72P

Chapter 18.2, Problem 18.74P

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