(a) Explanation Given information: The mass of the circular plate ( m ) is 4 kg. The impulsive force ( F Δ t ) is − ( 2.4 N ⋅ s ) k . The radius of the circle ( r ) is 180 mm. The distance between the BG ( b ) is 150 mm. Calculation: Calculate the moment of inertia ( I x ) of plate about x axis using the formula: I x = 2 ( 1 4 m r 2 + m b 2 ) Substitute 4 kg for m , 180 mm for r , and 150 mm for b . I x = 2 ( 1 4 × 4 ( 180mm× 1m 1000mm ) 2 + 4 ( 150mm× 1m 1000mm ) 2 ) = 0.2448 kg ⋅ m 2 Calculate the moment of inertia ( I y ) of plate about y axis using the formula: I y = 2 ( 1 2 m r 2 + m r 2 ) Substitute 4 kg for m and 180 mm for r . I y = 2 ( 1 2 × 4 ( 180mm × 1m 1000mm ) 2 + 4 ( 180mm × 1m 1000mm ) 2 ) = 0.3888 kg ⋅ m 2 Calculate the moment of inertia ( I z ) of plate about z axis using the formula: I z = 2 ( 1 4 m r 2 + m ( b 2 + r 2 ) ) Substitute 4 kg for m , 180 mm for r , and 150 mm for b . I z = 2 ( 1 4 × 4 ( 180mm× 1m 1000mm ) 2 + 4 [ ( 180m 1000 ) 2 + ( 150 m 1000 ) 2 ] ) = 0.504 kg ⋅ m 2 Calculate the moment of inertia ( I x y ) of plate about xy axis using the formula: I x y = m b ( − r ) + m ( − b ) r = − m b r − m b r = − 2 m b r Substitute 4 kg for m , 180 mm for r , and 150 mm for b . I x y = − 2 × 4 × ( 180mm× 1m 1000mm ) × ( 150mm× 1m 1000mm ) = − 0.216 kg ⋅ m 2 The moment of inertia about xz axis ( I x z ) and yz axis ( I y z ) are zero respectively (b) To determine The angular velocity ( ω ) of the assembly.

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Free Damped Vibrations

Free vibration is a type of vibration in which the external force is removed after giving the initial displacement to a body/system. In other words, in this type of vibration, force is applied at the initial displacement of the system, and after the init…

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Chapter 18.1, Problem 18.28P

(a)

To determine

The velocity (v¯) of the mass center G of the assembly.

(b)

To determine

The angular velocity (ω) of the assembly.

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Chapter 18.1, Problem 18.27P

Chapter 18.1, Problem 18.29P

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The velocity ( v ¯ ) of the mass center G of the assembly.

Solution Summary: The author calculates the moment of inertia of plate about x axis using the formula: I_x=2(14m

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The velocity (v¯) of the mass center G of the assembly.

The angular velocity (ω) of the assembly.

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Chapter 18 Solutions