In a damped harmonic oscillator, if the mass is doubled while keeping the damping coefficient constant, what happens to the damping ratio?

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Q: In a damped harmonic oscillator, if the mass is doubled while keeping the damping coefficient constant, what happens to the damping ratio?

Solution: Damping ratio (ζ) = c / (2√(mk)). If m is doubled, ζ is halved.

Steps: 7

Step 1: Understand the formula for the damping ratio (ζ), which is ζ = c / (2√(mk)). Here, c is the damping coefficient, m is the mass, and k is the spring constant.
Step 2: Identify that in this scenario, the mass (m) is being doubled. So, if the original mass is m, the new mass will be 2m.
Step 3: Substitute the new mass into the damping ratio formula. The new damping ratio becomes ζ' = c / (2√(2m * k)).
Step 4: Simplify the new damping ratio. This can be rewritten as ζ' = c / (2√(2) * √(m * k)).
Step 5: Notice that √(2) is a constant. Therefore, the new damping ratio ζ' is equal to the original damping ratio ζ divided by √(2).
Step 6: Since √(2) is approximately 1.414, this means that the new damping ratio ζ' is less than the original damping ratio ζ, specifically ζ' = ζ / √(2).
Step 7: Conclude that if the mass is doubled while keeping the damping coefficient constant, the damping ratio is halved.