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A slender uniform rigid bar of mass m is hinged at O and supported by two spr i ngs, wit h stiffnesses 3k and k, and a damper with damping coefficient c, as shown in the figure. For the system to be critically damped, t he ratio c / √km should be
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- 2
- 2 √7
- 4
- 4 √7
- 2
Correct Option: D
| I0 = | L2 | + m |  |  | 2 | |
| 12 | 4 |
| = | + | = | mL2 | |||
| 12 | 16 | 48 |
Torque about hinge ‘O’
| = | θ̇̇ + 3k. | | | θ + k. | θ + C. | θ̇ = 0 | ||||
| 48 | 16 | 16 | 16 |
| ⇒ | θ̇̇ + | θ + | θ̇ = 0 | |||
| 48 | 16 | 16 |
| 2ξωn = | = | ||
| 16 × 7 m | 7m |
| ⇒ 2√km = | × | ||
| 7 | 6 |
| = 4√7 | ||
| √km |
