Question

A force of 400 Newtons stretches a spring 2 meters. A mass of 50 kilograms is attached to the end of the spring and is initially released from the equilibrium position with an upward velocity of 10 m/s. After finding the equation of motion, calculate x(t = pi/12).

Answer 1

If we start taking into account the free body diagram, we then start damping motion of a body by means of this equation:
m * d^2 x/dt^2 + c* dx/dt + k* x = 0

free undamped motion is when (c =0)
therefore
50* d^2 x/dt^2 + 200* x = 0
w = omega = sqrt(k/m) = sqrt(200/50) = 2

Then

A*cos(w*t) + B*sin(w*t) = x

find A and B using intial conditions and you will get the rest that you need. Hope this helps