Question

Aspring mass system is characterized by k=16Nm/sec and m=1.0kg .the system is oscillating with an amplitude of 0.20m then i) calculate the angular velocity of oscillation.ii)obtain an expression for the velocity of the block as a function of displacement and calculate its value
at X=0.1m .iii)calculate energy of the spring mass system

Answer 1

A spring mass system oscillates in a simple harmonic motion with an angular speed  ω.  The restoration force F due to the spring on the mass m is given by:

               F = m d²x/dt² = - k x
                d²x /dt² = - (k/m) x    = - ω² x
              ω = √(k/m)  = √16/1 = 4 rad/sec
                             as k = 16 N m/sec  and  m = 1.0 kg

  Amplitude = A = 0.20 meters
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The displacement x from its mean position at any point of time t of the mass m can be given as:  
       x = A Sin (ω t)  = 0.20 Sin 4 t

   velocity instantaneous  of the mass = v = dx/dt =
       v = A ω Cos ωt        obtained by differentiating x wrt t.
         = ω √[A² - x²]
    
   So  v = 4 * √[ 0.2² - 0.1²]  = 0.4 * √3 m/sec
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Energy of the spring mass system is the total potential energy when the displacement is maximum ie., displacement = amplitude  or  the total KE when the displacement is 0.

Total energy  =  1/2 k A²  = 1/2 * 16 * 0.2² = 0.32 Joules.

we can also calculate this as follows:
     The velocity of the mass when x = 0 is:  v = ω A
     So  total energy = 1/2 * m * v² = 1/2 * 1 * (4 * 0.2)² = 0.32 Joules