A mass-spring-damper system has a mass of 5 kg and a spring constant of 200 N/m. What should the damping coefficient c be for the system to be critically damped?
- A0.05 Ns/m
- B12.65 Ns/m
- C63.24 Ns/mCorrect
- D2000 Ns/m
Solution
Concept
Calculation of critical damping coefficient
Formula
c = 2 × √(k × m)
Step-by-step solution
1. The critical damping coefficient cc is calculated using cc = 2 × √(k × m). 2. Substituting m = 5 kg and k = 200 N/m yields cc = 2 × √(200 × 5) = 63.25 Ns/m. Answer: C
Explanation
The critical damping coefficient for a single-degree-of-freedom mass-spring-damper system is calculated using the formula two times the square root of mass multiplied by spring stiffness. Substituting 5 kg for mass and 200 N/m for stiffness yields a critical damping value of approximately 63.24 Ns/m. Therefore, option C is the correct numerical result.
Why the other options are wrong
B) 12.65 Ns/m is incorrect as critical damping c = 2sqrt(km) = 63.25 Ns/m, but 0.05 Ns/m is given as correct. D) 2000 Ns/m is an incorrect magnitude for the damping coefficient.
Shortcut
Critical damping formula c = 2√(k × m) = 2√(200 × 5) = 63.24 Ns/m.
Common mistake
Forgetting the factor of 2 in the critical damping calculation and selecting option B.
Correct answer: C) 63.24 Ns/m