Critical damping coefficient — a 1 kg mass is attached to a spring of stiffness 0.7 N/mm. What is the critical damping coefficient c_c for this system?

Introduction / Context: The critical damping coefficient defines the threshold between oscillatory (underdamped) and non-oscillatory (overdamped) responses in a mass–spring–damper system. It is a key design target when fast return without overshoot is desired (e.g., measuring instruments).

Given Data / Assumptions:

• Mass m = 1 kg.
• Stiffness k = 0.7 N/mm = 700 N/m.
• Viscous damping model; find c_c.

Concept / Approach: For a single-DOF system, c_c=2√(km). Substituting k and m gives the required value. Units must be consistent (N·s/m).

Step-by-Step Solution:

Verification / Alternative check: Natural frequency ω_n=√(k/m)=√700≈26.46 rad/s. Since c_c=2mω_n=2×1×26.46≈52.92 N·s/m, which matches.

Why Other Options Are Wrong:
1.4 N·s/m — far below critical; implies very light damping.
18.52 N·s/m — still underdamped relative to c_c.
529.2 N·s/m — ten times the correct value; overdamped by a wide margin.

Common Pitfalls: Forgetting to convert N/mm to N/m; using c=2k instead of c=2√(km).

Final Answer: 52.92 N-s/m.