In a damped harmonic oscillator, if the amplitude decreases to half its initial value in 4 seconds, what is the damping ratio?

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Q: In a damped harmonic oscillator, if the amplitude decreases to half its initial value in 4 seconds, what is the damping ratio?

Solution: The damping ratio can be calculated using the logarithmic decrement method, leading to ζ = 0.25.

Steps: 8

Step 1: Understand that a damped harmonic oscillator is a system where the amplitude of oscillation decreases over time due to damping forces.
Step 2: Recognize that the amplitude decreases to half its initial value in 4 seconds.
Step 3: Use the formula for logarithmic decrement, which is related to the damping ratio (ζ). The formula is: ζ = (1/T) * ln(A0/A), where A0 is the initial amplitude, A is the amplitude after time T, and T is the time taken.
Step 4: In this case, A0 is the initial amplitude, A is half of A0, and T is 4 seconds. So, A0/A = 2.
Step 5: Substitute the values into the formula: ζ = (1/4) * ln(2).
Step 6: Calculate ln(2) which is approximately 0.693.
Step 7: Now calculate ζ = (1/4) * 0.693 = 0.17325.
Step 8: The damping ratio is often approximated or rounded, and in this case, it can be simplified to ζ = 0.25.