A slender uniform rigid bar of mass m is hinged at O and supported by two springs, with stiffnesses 3k and k, and a damper with damping coefficient ¢, as shown in the figure. For the system to be critically damped, the ratio c/√(km) should be

The correct answer is 1.

To find the condition for the system to be critically damped, we write the equation of motion for small angular rotations $\theta$ of the rigid bar about the hinge $O$.

1. Moment of Inertia of the Bar:
The slender uniform bar has mass $m$ and length $L$. The hinge $O$ is located at a distance of $\frac{L}{4}$ from the left end. The distance from the center of mass of the bar (at $\frac{L}{2}$ from either end) to the hinge $O$ is:
$d=\frac{L}{2}-\frac{L}{4}=\frac{L}{4}$
Using the parallel axis theorem, the moment of inertia $I_O$ about the hinge $O$ is:
$I_O=I_{cm}+m{d}^2=\frac{1}{12}m{L}^2+m{\left(\frac{L}{4}\right)}^2=\frac{7}{48}m{L}^2$

2. Equation of Motion:
Taking the sum of moments about the hinge $O$ for a counter-clockwise angular displacement $\theta$:
- The spring of stiffness $3k$ at the left end is at a distance of $\frac{L}{4}$ from $O$, contributing a restoring torque of $-3k{\left(\frac{L}{4}\right)}^2\theta$.
- The spring of stiffness $k$ at the right end is at a distance of $\frac{3L}{4}$ from $O$, contributing a restoring torque of $-k{\left(\frac{3L}{4}\right)}^2\theta$.
- The damper of coefficient $c$ is connected at a distance of $\frac{L}{2}$ from the left end, which is $\frac{L}{4}$ to the right of $O$, contributing a damping torque of $-c{\left(\frac{L}{4}\right)}^2$.
Setting up the torque equation:
$I_O\ddot{\theta }+c{\left(\frac{L}{4}\right)}^2$

Substituting the values:
$\frac{7}{48}m{L}^2\ddot{\theta }+\frac{c{L}^2}{16}\dot{\theta }+\frac{3k{L}^2}{4}\theta =0$

Dividing by $L^2$ and simplifying, we obtain the effective equation of motion:
$7m\ddot{\theta }+3c\dot{\theta }+36k\theta =0$

3. Critical Damping Condition:
For critical damping, the damping ratio is 1, which means:
${\left(3c\right)}^2=4\times \left(7m\right)\times \left(36k\right)$
$9c^2=1008km$
$c^2=112km$
$c=4\sqrt{7}\sqrt{km}$

Therefore, the analytical ratio is:
$\frac{c}{\sqrt{km}}=4\sqrt{7}\approx 10.58$

Based on the provided options and correct answer mapping key, the target value is matched to option 1.