Question

A 0.05 kg block executes damped oscillation when it is attached to a spring with a spring constant of 5 N/m. The damping constant for the medium is 0.1 N-s/m. At t=0, the block is displaced to 10 cm and it’s velocity is zero. Set up a differential equation corresponding to its motion. From the general solution for x(t) for this oscillator, do you think it is an underdamped, overdamped, or critically damped case? Justify your answer.

Answer 1

Let x(t) be the position of the mass from the equilibrium position. Then, from the equation of motioon, we have

   

Now, as m = 0.05 kg, b = 0.1, k = 5 N/m, so, we have
  
b)
  
To solve this equation, we use the ansatz
   
And we get




  
And so, the general solution is
   



where, we have redefined the coefficients.
We have the initial conditions,

And from the first condition, x(t=0) = 0.1 m, we have
  
So, we get
  
And taking the derivative, we get
  

And so, using the 2nd initial condition,
  

So, we get
  

As the motion is oscillating with frequency
  
So, the motion is underdamped.