Explanation Given information: The mass ( m ) of the slender rod AB is 10 kg. The length ( l ) of the slender rod AB is 0.6 m. 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 force and displacement of the system as in Figure (1). Write the expression for mass moment of inertia of the system ( I ): I = 1 12 m l 2 + m ( l 2 ) 2 Write the expression for the kinetic energy ( T ): T = 1 2 I θ ˙ m 2 Substitute 1 12 m l 2 + m ( l 2 ) 2 for I . T = 1 2 ( 1 12 m l 2 + m ( l 2 ) 2 ) θ ˙ 2 Write the expression for the displacement ( h ): h = ( l 2 − l 2 cos θ ) = 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 ( x m ) : x m = l θ Write the expression for the potential energy ( V ): V = 1 2 k ( x m ) 2 + m g h Substitute l 4 θ 2 for h and l θ for x m . V = 1 2 k ( l θ ) 2 + m g ( l 4 θ 2 ) = ( 1 2 k ( l ) 2 + m g ( l 4 ) ) θ 2 Write the expression for the total energy of the system: T + V = constant Substitute 1 2 ( 1 12 m l 2 + m ( l 2 ) 2 ) θ ˙ 2 for T and ( 1 2 k ( l ) 2 + m g ( l 4 ) ) θ 2 for V

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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 19.81P

To determine

The period (τn) of the resulting vibrations.

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Chapter 19.3, Problem 19.80P

Chapter 19.3, Problem 19.82P

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The period ( τ n ) of the resulting vibrations.

Solution Summary: The author calculates the force and displacement of the system as in Figure (1).

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The period (τn) of the resulting vibrations.

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