Explanation Given information: The mass of the plate is 8 kg and the angular velocity of the system is 6 rad / s . The figure-(1) represent the plates and the vertical shaft A B . Figure-(1) Write the expression of area of the plate. A = 1 2 b a ....... (I) Here, the base of the triangular plate is b and the height of the triangular plate is a . Write the expression of mass density of plate. ρ = m A ...... (II) Here, the mass of the plate is m . Substitute 1 2 b a for A in Equation (II). ρ = m ( 1 2 b a ) = 2 m b a ....... (III) Consider that he horizontal strip is a rectangular element. Write the expression of centroid along x direction. x c = x 2 ...... (IV) Here, the length of the elemental strip is x . Write the expression of centroid along x direction. y c = y ...... (V) Here, the width of the elemental strip is y . Write the expression of elemental area of the strip. d A = x d y ...... (VI) Here, the elemental width is d y . Write the expression of product moment of inertia about the coordinate axis. d I x y = x c y c d A ....... (VII) Substitute x 2 for x c , y for y c and x d y for d A in Equation (VII). d I x y = ( x 2 ) y x d y = ( x 2 y 2 ) d y ....... (VIII) Write the straight line equation of the plate. x b + y a = 1 x b = 1 − y a x = b − b y a ....... (IX) Write the expression of product moment of inertia about x y plane. I x y = ∫ 0 a 2 d I x y ...... (X) Substitute ( x 2 y 2 ) d y for d I x y in Equation (X). I x y = ∫ 0 a 2 { ( x 2 y 2 ) d y } = ∫ 0 a x 2 y d y ....... (XI) Substitute b − b y a for x in Equation (XI). I x y = ∫ 0 a ( b − b y a ) 2 y d y = ∫ 0 a ( b 2 y + b 2 y 3 a 2 − 2 b 2 y 2 a ) d y = b 2 [ y 2 2 ] 0 a + b 2 a 2 [ y 4 4 ] 0 a − 2 b 2 a [ y 3 3 ] 0 a = b 2 [ a 2 2 + a 4 4 a 2 − 2 a 3 3 a ] I x y = b 2 a 2 [ 1 2 + 1 4 − 2 3 ] = 1 12 a 2 b 2 Here, the product of moment of inertia about y z plane is zero because the plates lie in x y plane and the product of moment of inertia of body about the plane perpendicular to the plane of symmetry is zero. I y z = 0 Write the expression of moment of inertia of system about y axis. I y = 2 I plate ...... (XI) Here, the moment of inertia of plate is I plate . Write the expression of moment of inertia of plate. I plate = 1 12 a b 3 ...... (XII) Substitute 1 12 a b 3 for I plate in Equation (XI). I y = 2 ( 1 12 a b 3 ) = 1 6 a b 3 ....... (XIII) Write the expression of mass moment of inertia about x y plane. I ¯ x y = ρ I x y ....... (XIV) Substitute ( 2 m a b ) for ρ and 1 12 a 2 b 2 for I x y in Equation (XIV). I ¯ x y = ( 2 m a b ) ( 1 12 a 2 b 2 ) = 1 6 m a 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 18.1, Problem 18.19P

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The angular momentum about point G.

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The angular momentum about point G .

Solution Summary: The author explains the angular momentum about point G and the vertical shaft AB.

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