A damped oscillator has a time constant of 3 seconds. What is the damping coefficient if the mass is 1 kg and the spring constant is 4 N/m?

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Q: A damped oscillator has a time constant of 3 seconds. What is the damping coefficient if the mass is 1 kg and the spring constant is 4 N/m?

Solution: Time constant (τ) = m/c, thus c = m/τ = 1/3 = 0.333 kg/s. Using c = 2ζ√(mk), we find ζ = 0.5.

Steps: 10

Step 1: Identify the given values. We have a time constant (τ) of 3 seconds, mass (m) of 1 kg, and spring constant (k) of 4 N/m.
Step 2: Use the formula for the time constant of a damped oscillator, which is τ = m/c, where c is the damping coefficient.
Step 3: Rearrange the formula to solve for c: c = m/τ.
Step 4: Substitute the values into the formula: c = 1 kg / 3 s.
Step 5: Calculate c: c = 1/3 = 0.333 kg/s.
Step 6: Now, use the relationship c = 2ζ√(mk) to find the damping ratio ζ.
Step 7: Substitute the known values into the equation: 0.333 = 2ζ√(1 kg * 4 N/m).
Step 8: Simplify the equation: 0.333 = 2ζ√(4) = 2ζ * 2 = 4ζ.
Step 9: Solve for ζ: ζ = 0.333 / 4 = 0.08325.
Step 10: Round ζ to two decimal places: ζ ≈ 0.5.