GATE 2021 AE – Question 29
A single degree of freedom spring-mass-damper system is designed to return to its original unperturbed position in minimum possible time without overshooting. If mass is 10 kg, spring stiffness is 17400 N/m and natural frequency is 13.2 rad/s, the coefficient of damping is ___ N s/m (nearest integer).
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Correct answer: 264 to 264 OR 833 to 835
Explanation
The official key accepts either 264 to 264 OR 833 to 835. These are separate accepted answers, not one continuous interval.
To return to the original position as fast as possible **without overshooting**, the system must be **critically damped** ($\zeta=1$). A smaller damping ratio overshoots and oscillates, and a larger one is slower (over-damped).
**Critical damping coefficient:**
$$c_c=2\sqrt{km}=2\sqrt{17\,400\times10}=2\sqrt{174\,000}=2\times417.1=\mathbf{834\ Ns/m}.$$
(The formula uses $k$ and $m$ only; the quoted natural frequency is not needed.)
Official GATE 2021 answer key: https://gate2026.iitg.ac.in/doc/download/Answer_keys2021/ae_2021.pdf