Question

A spring mass system is characterized by k=16 N/m and m=1.0 kg. The system is oscillating with an amplitude of 0.20 m

1. Calculate the angular frequency of oscillation.
2. Obtain an expression for the velocity of the block as a function of displacement and calculate its value at x=0.1 m
3 . Also calculate energy of the spring mass system.

Answer 1

Spring mass system:
     k = 16 N/m    and  m = 1.0 kg
   Amplitude A = 0.20 m
    Let x be the compression of the spring at a time instant t.  x is the displacement of mass m from its mean position.

     F = restoration force on the mass exerted by the spring = - k x
       = m a  = equation of motion of the mass 
       =  m (- ω² x ) = - m ω² x                for a Simple harmonic motion.
   =>  ω = √(k/m)
         angular frequency = ω = √(16/1.0) = 4 rad/sec
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2.    total energy of the mass is conserved.
         E = elastic PE + KE = 1/2 k x² + 1/2 m v²
    when x = A = amplitude, velocity is 0  (at the extreme position of oscillation).
         E = 1/2 k A² = 1/2 * 16 * 0.20² = 0.32 Joules
      
   hence,  the expression :  1/2 m v² = 0.32 - 1/2 k x²
                         v² = 0.64 - 16 x²
                         v = √[0.64 - 16 x² ]
     v at x = 0.1 m is    √(0.64 - 0.16) = 0.6928 m/sec
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3.    The energy = 1/2 k A² = 0.32 Joules    calculated above in part (2)