A single degree of freedom system with mass $m$, damping coefficient ($c$), and stiffness ($k$) is subjected to a harmonic force $F(t) = F_0 \cos(\omega t)$.
1. Derive the equation of motion for the system.
2. Assuming the solution is of the form $x(t) = X \cos(\omega t - \phi)$, determine expressions for the amplitude ($X$) and phase angle ($\phi$) in terms of ($F_0$), ($k$), ($c$), ($\omega$), and ($m$).
3. Define the concept of resonance and explain how the damping ratio $\zeta = \frac{c}{2\sqrt{mk}}$ influences the resonance amplitude and frequency.
4. If the system is critically damped, explain the behavior of the system under forced vibration conditions and compare it to the underdamped and overdamped cases.
5. For a system with $m = 2 Kg.$, $k = 500 \frac{N}{m}$, $c = 10 \frac{Ns}{m}$, calculate the amplitude of steady-state vibration when the excitation frequency $\omega = 5 \frac{rad}{s}$, And $F_0 = 100 N$.
