Explanation Given information: The mass of the solid rectangular parallelepiped is m . The side of the square base of solid rectangular parallelepiped is a . The length of solid rectangular parallelepiped is 2 a . The angular velocity of solid rectangular parallelepiped about its diagonal A C ′ is ω . Calculation: Calculate the largest diagonal length ( d ) of the solid rectangular parallelepiped using the relation: d = a 2 + ( 2 a ) 2 + a 2 = a 2 + 4 a 2 + a 2 = 6 a 2 = a 6 Calculate the angular velocity ( ω ) of the system using the relation: ω = ω d ( − a i + 2 a j − a k ) Substitute a 6 for d . ω = ω a 6 ( − a i + 2 a j − a k ) = − ω a 6 a i + ω a 6 2 a j − ω a 6 a k = − ω 6 i + 2 ω 6 j − ω 6 k Write the value of ω x as follows: ω x = ω 6 Here, ω x is the angular velocity in x direction. Write the value of ω y as follows: ω y = 2 ω 6 Here, ω y is the angular velocity in y direction. Write the value of ω z as follows: ω z = ω 6 Here, ω z is the angular velocity in z direction. Write the expression for the mass per square plate ( m ′ ) as follows: m ′ = 1 10 m Here, m is the mass. Calculate moment of inertia in the x direction for each square plate ( I x ) using the formula: I x = 1 12 m ′ a 2 + m ′ a 2 = m ′ a 2 + 12 m ′ a 2 12 = 13 12 m ′ a 2 Substitute 1 10 m for m ′ . I x = 13 120 m a 2 Calculate moment of inertia in the y direction for each plate ( I y ) using the formula: I y = 1 6 m ′ a 2 Substitute 1 10 m for m ′ . I x = 1 60 m a 2 Calculate the moment of inertia in the z direction for each square plate ( I z ) using the formula: I z = 1 12 m ′ a 2 + m ′ a 2 = m ′ a 2 + 12 m ′ a 2 12 = 13 12 m ′ a 2 Substitute 1 10 m for m ′ . I z = 13 120 m a 2 Write the expression for the mass per square plate parallel to yz plane ( m ″ ) as follows: m ″ = 1 5 m Calculate the moment of inertia in the x direction for each square plate parallel to yz plane ( I x 1 ) using the relation: I x 1 = 1 12 m ″ ( a 2 + ( 2 a ) 2 ) = 5 12 m ″ a 2 Substitute 1 5 m for m ″ . I x 1 = 5 12 ( m 5 ) a 2 = 1 12 m a 2 Calculate the moment of inertia in the y direction for each square plate parallel to yz plane ( I y 1 ) using the relation: I y 1 = 1 12 m ″ a 2 + m ″ ( a 2 ) 2 = 1 12 m ″ a 2 + m ″ a 2 4 = 4 12 m ″ a 2 = 1 3 m ″ a 2 Substitute 1 5 m for m ″ . I y 1 = 1 3 ( m 5 ) a 2 = 1 15 m a 2 Calculate moment of inertia in the z direction for each square plate parallel to yz plane ( I z 1 ) using the relation: I z 1 = 1 12 m ″ ( 2 a ) 2 + m ″ ( a 2 ) 2 = 1 3 m ″ a 2 + 1 4 m ″ a 2 = 7 12 m ″ a 2 Substitute 1 5 m for m ″ . I z 1 = 7 12 ( m 5 ) a 2 = 7 60 m a 2 The mass for each plate parallel to xy plane is the same as that for yz plane. Calculate the moment of inertia in the x direction for each square plate parallel to xy plane ( I x 2 ) using the relation: I x 2 = 1 12 m ″ ( 2 a ) 2 + m ″ ( a 2 ) 2 = 1 3 m ″ a 2 + 1 4 m ″ a 2 = 7 12 m ″ a 2 Substitute 1 5 m for m ″ . I x 2 = 7 12 ( m 5 ) a 2 = 7 60 m a 2 Calculate moment of inertia in the y direction for each square plate parallel to xy plane ( I y 2 ) using the relation: I y 2 = 1 12 m ″ a 2 + m ″ ( a 2 ) 2 = 1 12 m ″ a 2 + m ″ a 2 4 = 4 12 m ″ a 2 = 1 3 m ″ a 2 Substitute 1 5 m for m ″

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 18.1, Problem 18.45P

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The kinetic energy (T) of the hollow parallelepiped.

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Chapter 18.1, Problem 18.44P

Chapter 18.1, Problem 18.46P

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The kinetic energy ( T ) of the hollow parallelepiped.

Solution Summary: The author calculates the kinetic energy ( T ) of the hollow parallelepiped using the relation: cd =sqrt6a

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The kinetic energy (T) of the hollow parallelepiped.

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Chapter 18 Solutions