A damped harmonic oscillator has a mass of 2 kg and a damping coefficient of 0.5 kg/s. What is the damping ratio if the spring constant is 8 N/m?

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Q: A damped harmonic oscillator has a mass of 2 kg and a damping coefficient of 0.5 kg/s. What is the damping ratio if the spring constant is 8 N/m?

Solution: Damping ratio (ζ) = c / (2√(mk)) = 0.5 / (2√(2*8)) = 0.5 / (2√16) = 0.5 / 8 = 0.0625.

Steps: 7

Step 1: Identify the given values: mass (m) = 2 kg, damping coefficient (c) = 0.5 kg/s, spring constant (k) = 8 N/m.
Step 2: Use the formula for the damping ratio (ζ): ζ = c / (2√(mk)).
Step 3: Calculate the product of mass (m) and spring constant (k): mk = 2 kg * 8 N/m = 16 kg·N/m.
Step 4: Find the square root of mk: √(mk) = √(16) = 4.
Step 5: Multiply the square root by 2: 2√(mk) = 2 * 4 = 8.
Step 6: Substitute the values into the damping ratio formula: ζ = 0.5 / 8.
Step 7: Calculate the damping ratio: ζ = 0.0625.