A damped oscillator has a time constant of 3 seconds. What is the amplitude afte

A damped oscillator has a time constant of 3 seconds. What is the amplitude after 6 seconds if the initial amplitude is 10 m?

A damped oscillator has a time constant of 3 seconds. What is the amplitude after 6 seconds if the initial amplitude is 10 m?

Step 1: Identify the initial amplitude (A0). In this case, A0 = 10 m.
Step 2: Identify the time constant (τ). Here, τ = 3 seconds.
Step 3: Identify the time (t) for which we want to find the amplitude. Here, t = 6 seconds.
Step 4: Use the formula for the amplitude of a damped oscillator: Amplitude after time t = A0 * e^(-t/τ).
Step 5: Substitute the values into the formula: Amplitude after 6 seconds = 10 * e^(-6/3).
Step 6: Simplify the exponent: -6/3 = -2.
Step 7: Now the formula looks like this: Amplitude after 6 seconds = 10 * e^(-2).
Step 8: Calculate e^(-2). The approximate value of e^(-2) is about 0.1353.
Step 9: Multiply 10 by 0.1353 to find the final amplitude: 10 * 0.1353 ≈ 1.353 m.
Step 10: Therefore, the amplitude after 6 seconds is approximately 1.353 m.

Damped Oscillator – A system where the amplitude of oscillation decreases over time due to energy loss, characterized by a time constant (τ).
Exponential Decay – The process by which the amplitude decreases exponentially over time, represented by the formula A(t) = A0 * e^(-t/τ).