A single-degree-of-freedom system has a mass of 1000 kg and a stiffness of 40,000 N/m. What is its natural frequency in Hz? (Use π ≈ 3.14)
- A1 HzCorrect
- B2 Hz
- C3 Hz
- D4 Hz
Solution
Concept
Natural frequency of single-degree-of-freedom system
Formula
f = (1 / (2 × π)) × √(k / m)
Step-by-step solution
1. The natural frequency in Hz is given by f = (1 / 2π) × √(k / m). 2. Substitute m = 1000 kg and k = 40,000 N/m into the frequency relationship. Answer: A
Explanation
The natural frequency in Hz is calculated using the formula fn = (1 / 2π) × √(k / m). Substituting a stiffness of 40,000 N/m and a mass of 1000 kg gives fn = (1 / 6.28) × √(40), which evaluates to approximately 1 Hz. Therefore, option A is correct.
Why the other options are wrong
C) 3 Hz overestimates the system's natural frequency derived from the given stiffness and mass. D) 4 Hz significantly exceeds the calculated frequency for a stiffness of 40,000 N/m and mass of 1000 kg.
Shortcut
Natural frequency fn = (1 / 2π) × √(k / m) = (1 / 6.28) × √(40) ≈ 1 Hz.
Common mistake
Expressing natural frequency in radians per second instead of dividing by 2π to convert the result to Hertz.
Correct answer: A) 1 Hz