A body of mass m is subject to an initial displacement $u_0$ and an initial velocity $v_0$ at time t = 0.
Assume there is no friction between the mass and the plane that it slides on. The mass is
connected to a spring k, a viscous damper c, and undergoes free vibration (P(t) = 0).
a) Draw the free body diagram of the system in Figure 2 showing all relevant forces for free
vibration.
b) Derive the equation of motion of the system
c) Find the general equation for displacement u(t) and coefficients $A_1$ and $A_2$ in terms of
the initial conditions u(0) = $u_0$ and v(0) = $v_0$, natural frequency, damped frequency,
and damping ratio. Show all relevant steps in finding the coefficients.
d) If m = 4 kg, c = 1 Ns/m, and k = 900 N/m, find the critical damping factor $C_c$. Is the
system underdamped, overdamped, or critically damped? Explain in a sentence the
relevance of the critical damping factor.
e) Calculate the displacement of the system for $u_0$ = 0, $v_0$ = 2 m/s, at time t = .003 s.
