Explanation Given information: The weight ( W d i s k ) of the semicircular disk is 6 lb. The weight ( W A B ) of the rod AB is 4 lb. The radius ( r ) of the semicircular plate is 6 in . . Assuming the acceleration due to gravity ( g ) as 32.2 ft/s 2 . Calculation: Show the position 1 and position 2 of the system with velocity as in the Figure (1). For position 1: Write the expression for the maximum velocity of disk ( v d i s k ) : The point of contact is an instantaneous center, v d i s k = ( r − c ) θ ˙ m Here, c is centroidal distance. Write the expression for centroidal distance c : c = 4 r 3 π Write the expression for the maximum velocity of rod ( v A B ) v A B = r θ ˙ m Calculate the mass of the disk ( m d i s k ) using the relation: m d i s k = W d i s k g Substitute 6 lb for W d i s k and 32.2 ft/s 2 for g . m d i s k = 6 32.2 = 0.186 lb ⋅ s 2 / ft Calculate the mass of the rod AB ( m A B ) using the relation: m A B = W A B g Substitute 4 lb for W A B and 32.2 ft/s 2 for g . m A B = 4 32.2 = 0.124 lb ⋅ s 2 / ft Write the expression for the moment of inertia ( I A ) of disk at A : I A = 1 2 m d i s k r 2 Calculate the centroidal the moment of inertia ( I ¯ D ) of disk: I ¯ D = I A − m d i s k c 2 Substitute 1 2 m d i s k r 2 for I A and 4 r 3 π for c . I ¯ D = 1 2 m d i s k r 2 − m d i s k ( 4 r 3 π ) 2 = m d i s k [ r 2 2 − 16 r 2 9 π 2 ] Write the expression for the kinetic energy ( T 1 ) : T 1 = 2 1 2 I ¯ D θ ˙ m 2 + 2 ( 1 2 ) m d i s k v d i s k 2 + 1 2 m A B v A B 2 Substitute m d i s k [ r 2 2 − 16 r 2 9 π 2 ] for I ¯ D , r θ ˙ m for v A B , 4 r 3 π for c , and ( r − c ) θ ˙ m for v d i s k . T 1 = 2 1 2 ( m d i s k [ r 2 2 − 16 r 2 9 π ] ) θ ˙ m 2 + 2 ( 1 2 ) m d i s k ( r − 4 r 3 π ) 2 θ ˙ m 2 + 1 2 m A B r 2 θ ˙ m 2 = [ ( 0.3199 + 0.3313 ) m d i s k r 2 + 0.5 m A B r 2 ] θ ˙ m 2 = ( 0.6512 m d i s k + 0

12th Edition

ISBN: 9781265566296

Author: BEER

Publisher: MCG

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Free undamped vibrations

Whenever an object/system displaces in either side of its equilibrium position with the help of an external/internal means, then the object/system moves regularly or randomly on both sides of its equilibrium position. This motion is called vibration. Gen…

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Chapter 19.3, Problem 19.91P

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The period (τn) of small oscillations of the system.

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Chapter 19.3, Problem 19.90P

Chapter 19.3, Problem 19.92P

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The period ( τ n ) of small oscillations of the system.

Solution Summary: The author shows the position 1 and position 2 of the system with velocity as in the Figure (1).

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The period (τn) of small oscillations of the system.

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