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In a damped harmonic oscillator, if the mass is doubled while keeping the dampin

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

Options:

  1. Doubles
  2. Halves
  3. Remains the same
  4. Increases by a factor of √2

Correct Answer: Halves

Solution: 

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

In a damped harmonic oscillator, if the mass is doubled while keeping the dampin

Practice Questions

Q1
In a damped harmonic oscillator, if the mass is doubled while keeping the damping coefficient constant, what happens to the damping ratio?
  1. Doubles
  2. Halves
  3. Remains the same
  4. Increases by a factor of √2

Questions & Step-by-Step Solutions

In a damped harmonic oscillator, if the mass is doubled while keeping the damping coefficient constant, what happens to the damping ratio?
  • 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.
  • Damped Harmonic Oscillator – A system where the motion is affected by a damping force, typically proportional to the velocity, leading to a gradual loss of energy.
  • Damping Ratio (ζ) – A dimensionless measure describing how oscillations in a system decay after a disturbance, defined as ζ = c / (2√(mk)).
  • Effect of Mass on Damping Ratio – Doubling the mass while keeping the damping coefficient constant affects the damping ratio, specifically halving it.
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