Explanation Given information: The mass ( m ) of the slender rod AB is 5 kg. The length ( l ) of the slender rod AB is 0.6 m. The mass ( m C ) of the both the collar is 2.5 kg. The spring constant ( k ) is 1.5 kN/m . Assume the acceleration due to gravity ( g ) as 9.81 m/s 2 . Calculation: Show the velocity in collar A and B as in Figure (1). Write the expression for mass moment of inertia rod ( I ): I = ( m l 2 3 ) Write the expression the velocity ( v A ) of collar A : v A = l θ ˙ Write the expression for the kinetic energy ( T ): T = 1 2 I θ ˙ 2 + 1 2 m C v A 2 Substitute ( m l 2 3 ) for I and l θ ˙ for v A . T = 1 2 ( m l 2 3 ) θ ˙ 2 + 1 2 m C ( l θ ˙ ) 2 = ( 1 2 ( m l 2 3 ) + 1 2 m C l 2 ) θ ˙ 2 Show the position of displacement as in the Figure (2). Write the expression for the displacement ( h ) of bar AB : h = ( l 2 − l 2 cos θ ) h = l 2 ( 1 − cos θ ) ( ∵ ( 1 − cos θ = 2 sin 2 θ 2 ) ) = l 2 2 sin 2 θ 2 For small oscillation 2 sin 2 θ 2 ≈ θ 2 2 . h = l 2 θ 2 2 = l 4 θ 2 Write the expression for the displacement ( h 1 ) of collar: h 1 = ( l − l cos θ ) = l ( 1 − cos θ ) ( ∵ ( 1 − cos θ = 2 sin 2 θ 2 ) ) = l ( 2 sin 2 θ 2 ) For small oscillation 2 sin 2 θ 2 ≈ θ 2 2 . h 1 = l θ 2 2 = l 2 θ 2 Write the expression for the displacement ( x m ) : x m = l θ Write the expression for the potential energy ( V ): V = 1 2 k ( x m ) 2 + m g h + m C g h 1 Substitute l 4 θ 2 for h , l θ for x m , and l 2 θ 2 for h 1 . V = 1 2 k ( l θ ) 2 + m g ( l 4 θ 2 ) + m C g ( l 2 θ 2 ) = ( k l 2 2 + m g l 4 + m C g l 2 ) θ 2 Write the expression for the total energy of the system: T + V = constant Substitute ( 1 2 ( m l 2 3 ) + 1 2 m C l 2 ) θ ˙ 2 for T and ( k l 2 2 + m g l 4 + m C g l 2 ) θ 2 for V

11th Edition

ISBN: 9781259633133

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 19.82P

To determine

The period (τn) of the resulting vibrations.

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Chapter 19.3, Problem 19.81P

Chapter 19.3, Problem 19.83P

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The period of vibration of the system shown is observed to be 0.6 s. After cylinder B has been removed, the period is observed to be 0.5 s. Determine (a) the weight of cylinder A(b) the constant of the spring.

A 5-kg collar Cis released from rest in the position shown and slides without friction on a vertical rod until it hits a spring with a constant of k= 720 N/m that it compresses. The velocity of the collar is reduced to zero, and the collar reverses the direction of its motion and returns to its initial position. The cycle is then repeated. Determine (a) the period of the motion of the collar, (b) the velocity of the collar 0.4 s after it was released. (Note: This is a periodic motion, but it is not simple harmonic motion.)

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Chapter 19 Solutions

VECTOR MECH...,STAT.+DYNA.(LL)-W/ACCESS

Ch. 19.1 - A particle moves in simple harmonic motion....Ch. 19.1 - Prob. 19.2P Ch. 19.1 - Prob. 19.3P Ch. 19.1 - Prob. 19.4P Ch. 19.1 - Prob. 19.5P Ch. 19.1 - Prob. 19.6P Ch. 19.1 - Prob. 19.7P Ch. 19.1 - Prob. 19.8P Ch. 19.1 - Prob. 19.9P Ch. 19.1 - Prob. 19.10P

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The period ( τ n ) of the resulting vibrations.

Solution Summary: The author shows the velocity in collar A and B as in Figure (1). The mass of the slender rod AB is 5 kg, and the length is 0.6 m.

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The period (τn) of the resulting vibrations.

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Chapter 19 Solutions