Explanation Given information: The base is a , the length is 2 a and the height is a . Write the expression of kinetic energy. T = 1 2 m v 2 + 1 2 I x ω x 2 + 1 2 I y ω y 2 + 1 2 I z ω z 2 ....... (I) Here, the mass of the disk is m , the velocity is v , the moment of inertia about x axis is I x , the moment of inertia about y axis is I y , the moment of inertia about z axis is I z ,the angular velocity about x axis is ω x , the angular velocity about y axis is ω y and the angular velocity about z axis is ω z . Write the expression of diagonal of the parallelepiped. d = l 2 + b 2 + h 2 ....... (II) Here, the of base the parallelepiped is l and the length of the parallelepiped is b and the height of the parallelepiped is h . Substitute 2 a for b , a for l and a for h in Equation (I): d = ( a ) 2 + ( 2 a ) 2 + ( a ) 2 = ( a ) 2 + 4 ( a ) 2 + ( a ) 2 = a 6 Write the expression of vector form of angular velocity of the disk. ω = ω d ( − l i + b j − h k ) ........ (III) Here, the length of the diagonal of the disk is d , the unit vector in x direction is i , the unit vector in y direction is j and the unit vector in z direction is k . Substitute 2 a for b , a for l , a for h and a 6 for d in Equation (II). ω = ω a 6 ( − a i + 2 a j − a k ) = a ω a 6 ( − i + 2 j − k ) = ω 6 ( − i + 2 j − k ) = − ( ω 6 ) i + 2 ( ω 6 ) j − ( ω 6 ) k ....... (IV) Write the expression of vector form of angular velocity of the disk. ω = ω x i + ω y j + ω x k ........ (V) Here, the x component of the angular velocity is ω x , the y component of the angular velocity is ω y and the z component of the angular velocity is ω z . Compare the unit vector i of Equation (III) and (IV). ω x = − ( ω 6 ) ....... (VI) Compare the unit vector j of Equation (III) and (IV). ω y = 2 ( ω 6 ) ....... (VII) Compare the unit vector k of Equation (III) and (IV). ω z = − ( ω 6 ) ....... (VIII) Write the expression of mass of the square plate. m ′ = 1 10 m ....... (IX) Here, the total mass of the parallelepiped is m . Write the expression of moment of inertia of the square plate about the x axis. I x 1 = 1 12 m ′ l 2 + m ′ l 2 ...... (X) Substitute a for l and 1 10 m for m ′ in Equation (X): I x 1 = 1 12 ( 1 10 m ) a 2 + ( 1 10 m ) a 2 = 1 120 m a 2 + 1 10 m a 2 = 13 120 m a 2 ....... (XI) Write the expression of moment of inertia of the square plate about the y axis. I y 1 = 1 6 m ′ l 2 ....... (XII) Substitute a for l and 1 10 m for m ′ in Equation (XII): I y 1 = 1 6 ( 1 10 m ) a 2 = 1 60 m a 2 ....... (XIII) Write the expression of moment of inertia of the square plate about the z axis. I z 1 = 1 12 m ′ l 2 + m ′ l 2 ...... (XIV) Substitute a for l and 1 10 m for m ′ in Equation (XIV): I z 1 = 1 12 ( 1 10 m ) a 2 + ( 1 10 m ) a 2 = 1 120 m a 2 + 1 10 m a 2 = 13 120 m a 2 ....... (XV) Write the expression of mass of the parallel plate. m ′ ′ = 1 5 m ....... (IX) Write the expression of moment of inertia of parallel plate about y z plane in x axis. I x 2 = 1 12 m ′ ′ ( l 2 + b 2 ) ........ (XVI) Substitute 2 a for b a for l and 1 5 m for m ′ ′ in Equation (XVI): I x 2 = 1 12 ( 1 5 m ) ( ( a ) 2 + ( 2 a ) 2 ) = 1 60 m ( ( a ) 2 + 4 ( a ) 2 ) = 1 12 m a 2 ....... (XVII) Write the expression of moment of inertia of parallel plate about y z plane in y axis. I y 2 = 1 12 m ′ ′ l 2 + m ′ ′ ( l 2 ) 2 ........ (XVIII) Substitute a for l and 1 5 m for m ′ ′ in Equation (XVIII): I y 2 = 1 12 ( 1 5 m ) a 2 + ( 1 5 m ) ( a 2 ) 2 = 1 5 m ( 1 12 a 2 + a 2 4 ) = 1 5 m ( 4 a 2 12 ) = 1 15 m a 2 ........ (XIX) Write the expression of moment of inertia of parallel plate about y z plane in z axis. I z 2 = 1 12 m b 2 + m ′ ′ ( l 2 ) 2 ........ (XX) Substitute 2 a for b a for l and 1 5 m for m ′ ′ in Equation (XX): I z 2 = 1 12 m ( 2 a ) 2 + ( 1 5 m ) ( a 2 ) 2 = 1 3 m a 2 + ( 1 5 m ) ( a 2 4 ) = 1 3 m a 2 + 1 20 m a 2 = 7 60 m a 2

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Chapter 18.1, Problem 18.45P

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The kinetic energy of the solid parallelepiped.

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Ch. 18.1 - Prob. 18.1P Ch. 18.1 - Prob. 18.2P Ch. 18.1 - Prob. 18.3P Ch. 18.1 - A homogeneous disk of weight W=6 lb rotates at the...Ch. 18.1 - Prob. 18.5P Ch. 18.1 - A solid rectangular parallelepiped of mass m has a...Ch. 18.1 - Solve Prob. 18.6, assuming that the solid...Ch. 18.1 - Prob. 18.8P Ch. 18.1 - Determine the angular momentum HD of the disk of...Ch. 18.1 - Prob. 18.10P

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