Question

A spring mass system is characterized by 1
16Nm−
k = and m = 0.1 kg.The system is
oscillating with an amplitude of 0.20 m. i) Calculate the angular frequency of oscillation.
ii) Obtain an expression for the velocity of the block as a function of displacement and
calculate its value at x = 1.0 m. iii) Also calculate energy of the spring-mass system.

Answer 1

  equation of motion and force:
           F = m a = m d²x/dt² = - k x
                     d²x/dt² = - k/m x

      let x = A Sin (ωt + Ф)
             then  d²x/dt² = - A ω² Sin(ωt+Ф) = - ω² x

       k = 116 N/m  or  16 N/m  ???      which one ?
       m = 0.1 kg
       A = 0.20 meters

SHM :  angular frequency = ω = √(k/m) = √(116/0.1) = 34 rad/sec
                         if  k = 16 N/m,    ω = √160 = 4√10 rad/sec
   ==========
     x = A Sin (ωt + Ф)          = displacement from the mean position
     v = velocity of the particle executing the SHM
     v = dx/dt = A ω Cos (ω t + Ф)
            v  = ω  √A² - x² )  = ω A  √[1 - x²/A² ]
====================
   x = 1.0 m      this value is not possible, as  amplitude is 0.20 m.  SO x has to be less than 0.2 m.  Is it 0.1 meters ?      x has to be less than or equal to amplitude.

   v = 4√10 rad/sec * √[0.2² - 0.1²] meters = 2.19 m/sec

========================

Energy of the spring mass system : 
      = 1/2 m v² at the mean equilibrium position , as x = 0  and PE = 0
         = 1/2 k A²  at the extreme position when x = A, as  v = 0 and KE = 0.
     = 1/2 * 0.1 kg * 0.20² = 0.002 Joules