The amplitude of an oscillator decreases to 36.2% of its initial value in 20.5 s. What is the value of the time constant?
.362=1*e^-kt
put in t=20.5sec, solve for k. Now some folks use the term time constant for 1/k
The time constant (τ) can be found using the formula:
τ = -t / ln(A / A0)
Where:
τ is the time constant,
t is the time taken for the amplitude to decrease to a certain value (36.2% in this case),
A is the final amplitude (36.2% of the initial value),
A0 is the initial amplitude.
Given that the amplitude decreases to 36.2% of its initial value in 20.5 seconds, we can substitute the values into the formula and calculate the time constant.
τ = -(20.5 s) / ln(0.362)
Using a scientific calculator,
τ ≈ -20.5 s / ln(0.362)
≈ -20.5 s / (-1.018)
Calculating further:
τ ≈ 20.07 s
Therefore, the value of the time constant is approximately 20.07 seconds.
To determine the value of the time constant, we need to use the equation for the decay of an oscillator's amplitude over time.
The equation is given by:
A = A0 * e^(-t/τ)
Where:
A is the final amplitude (in this case, 36.2% of the initial amplitude)
A0 is the initial amplitude
t is the time elapsed (20.5 s in this case)
τ (tau) is the time constant we want to find.
We can rearrange the equation to solve for τ:
ln(A/A0) = -t/τ
Now we can substitute the given values into the equation and solve for τ.
ln(0.362) = -20.5/τ
Using a scientific calculator, we can take the natural logarithm (ln) of 0.362 to get -1.018.
-1.018 = -20.5/τ
Now, we can solve for τ by rearranging the equation:
τ = -20.5 / -1.018
τ ≈ 20.14 seconds
Therefore, the value of the time constant is approximately 20.14 seconds.