The equation of motion of a spring-mass-damper system is given by
​$$\ddot{x} +4\dot{x} + 16x = 12sin 5t$$​​
What is the damping factor of the system?

Correct option is B

​$$\begin{aligned}&\text{We are given a standard second-order linear differential equation for a \textbf{spring-mass-damper system}:} \\&\ddot{x} + 2\zeta \omega_n \dot{x} + \omega_n^2 x = \frac{F(t)}{m} \\[1em]&\text{Compare this with the given equation:} \\&\ddot{x} + 4\dot{x} + 16x = 12\sin 5t \\[1em]&\text{From comparison:} \\&\bullet \quad 2\zeta \omega_n = 4 \\&\bullet \quad \omega_n^2 = 16 \Rightarrow \omega_n = \sqrt{16} = 4 \\[1em]&\text{Now plug into the damping ratio formula:} \\&\zeta = \frac{2\zeta \omega_n}{2 \omega_n} = \frac{4}{2 \cdot 4} = \frac{4}{8} = \boxed{0.5}\end{aligned}$$​​