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

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Author: BEER

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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 ( 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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