# A spring mass system is characterized by K=16N/m and m= 1.0kg. System is oscillate with an amplitude of 0.20m.(1)Calculate the angular frequency of oscillation (2) obtain an expression for the velocity of the block. As a function of displacement

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2015-04-14T18:53:41+05:30

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A spring mass system  is characterized by the equation of motion:

m a = F = m d² x / d t² = - k x
d² x/ d t² = - k/m  x  = - ω₀² x

the solution for displacement at time t is given by  :
x = A Cos (ω₀ t + Ф)

ω₀ is the natural frequency of the system for simple harmonic motion.
=  √(k/m) = √(16/1) = 4 rad /sec.
angular frequency is 4 rad /sec.

A = amplitude of the vibration = 0.20 metes.
Ф = initial phase of the system, given by

Cos Ф = x₀ / A ,  where x₀ = the position of mass from the mean position when the oscillation has started.

displacement x = A Cos (ω₀ t + Ф)
differentiate :  dx/dt = v = velocity of mass m at time t
v = - A * ω₀  sin (ω₀ t + Ф)
v = -  0.20 * 4 Sin (ω₀ t + Ф)

v = dx/dt = - 0.80 Sin 4 t      if  the initial phase = 0°.