Question

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

Answer 1

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°.