A damper with damping coefficient, \( c \), is attached to a mass of 5 kg and spring of stiffness 5 kN/m as shown in the figure. The system undergoes under-damped oscillations.
If the ratio of the 3rd amplitude to the 4th amplitude of oscillations is 1.5, the value of \( c \) is _________ Ns/m (rounded off to the nearest integer).
Show Hint
For underdamped systems, use the relationship between amplitude ratio and damping ratio to find the damping coefficient.
For an underdamped system, the amplitude of oscillation decreases over time, and the ratio of amplitudes is given by:
\[
\frac{A_3}{A_4} = \frac{1}{\left( 1 + 2 \zeta \right)^2},
\]
where \( \zeta \) is the damping ratio. The relationship between damping ratio and damping coefficient is given by:
\[
\zeta = \frac{c}{2 \sqrt{km}}.
\]
We are given that the ratio \( \frac{A_3}{A_4} = 1.5 \). Solving for \( c \) using the given values, we get:
\[
c \approx 19 \, \text{Ns/m}.
\]
Thus, the value of \( c \) is approximately \( 19 \, \text{Ns/m} \).
