The amplitude of an oscillator decreases to 36.8% of its initial value in 10.0 s. What is the value of the time constant?

Verified step by step guidance

1. Step 1: Recognize that the problem involves exponential decay, which is described by the equation \( A(t) = A_0 e^{-t/\tau} \), where \( A(t) \) is the amplitude at time \( t \), \( A_0 \) is the initial amplitude, and \( \tau \) is the time constant.
2. Step 2: Identify the given values: \( A(t) = 0.368 A_0 \) (36.8% of the initial amplitude), and \( t = 10.0 \, \text{s} \). The goal is to solve for \( \tau \), the time constant.
3. Step 3: Substitute the given values into the exponential decay formula: \( 0.368 A_0 = A_0 e^{-10.0/\tau} \). Cancel \( A_0 \) from both sides since it is non-zero.
4. Step 4: Simplify the equation to \( 0.368 = e^{-10.0/\tau} \). Take the natural logarithm (\( \ln \)) of both sides to isolate \( \tau \): \( \ln(0.368) = -10.0/\tau \).
5. Step 5: Rearrange the equation to solve for \( \tau \): \( \tau = -10.0 / \ln(0.368) \). This expression gives the time constant in terms of the given values.

Key Concepts

Amplitude and Damping

Amplitude refers to the maximum extent of a vibration or oscillation, measured from the position of equilibrium. In oscillatory systems, damping is the effect that reduces the amplitude over time, often due to energy loss from friction or resistance. Understanding how amplitude decreases is crucial for analyzing the behavior of oscillators.

Exponential Decay

Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value. In the context of oscillators, the amplitude decreases exponentially over time, which can be mathematically represented by the equation A(t) = A0 * e^(-t/τ), where A0 is the initial amplitude, τ is the time constant, and t is time.

Time Constant (τ)

The time constant, denoted as τ, is a key parameter in exponential decay processes. It represents the time required for the amplitude to decrease to approximately 36.8% of its initial value. In this scenario, knowing the time constant allows us to quantify how quickly the oscillator loses energy and how its motion is affected over time.