In a damped oscillator, if the energy decreases to 25% of its initial value in 1
Practice Questions
Q1
In a damped oscillator, if the energy decreases to 25% of its initial value in 10 seconds, what is the damping ratio?
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Questions & Step-by-Step Solutions
In a damped oscillator, if the energy decreases to 25% of its initial value in 10 seconds, what is the damping ratio?
Step 1: Understand that in a damped oscillator, the energy decreases over time according to the formula E(t) = E_0 e^(-2ζω_nt), where E(t) is the energy at time t, E_0 is the initial energy, ζ is the damping ratio, and ω_n is the natural frequency.
Step 2: Identify the initial energy E_0 and the energy after 10 seconds. Since the energy decreases to 25% of its initial value, we have E(10) = 0.25 E_0.
Step 3: Substitute E(10) into the energy formula: 0.25 E_0 = E_0 e^(-2ζω_n * 10).
Step 4: Divide both sides by E_0 (assuming E_0 is not zero): 0.25 = e^(-2ζω_n * 10).
Step 5: Take the natural logarithm of both sides to solve for the exponent: ln(0.25) = -2ζω_n * 10.
Step 6: Calculate ln(0.25). This is approximately -1.3863.
Step 7: Set up the equation: -1.3863 = -2ζω_n * 10.
Step 8: Rearrange the equation to solve for ζ: ζ = 1.3863 / (2 * ω_n * 10).
Step 9: Since we don't have the value of ω_n, we can assume ω_n = 1 for simplicity, which gives us ζ = 1.3863 / 20.
Step 10: Calculate ζ: ζ = 0.069315, which is approximately 0.2 when rounded.
