Given that the amplitude of an oscillator decreases to 36.7% of its initial value in 29.5 s, we want to get the time constant. We will get:
k = 0.041 s^-1
We can write a really simple oscillation equation as:
f(t) = A*cos(k*t)
Where A is the amplitude and k is the time constant.
We know that the amplitude decreases to 36.7% of its initial value to 29.5 seconds, this means that:
f(29.5s) = 0.367*A = A*cos(k*29.5s)
Dividing both sides by A, we get:
0.367 = cos(k*29.5s)
So we just need to solve the above equation for k.
If we use the inverse cosine function in both sides, we get:
Acos(0.367) = Acos(cos(k*29.5s))
Acos(0.367) = k*29.5s
1.195 = k*29.5s
1.195/29.5s = k = 0.041 s^-1
The time constant is:
k = 0.041 s^-1
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