The amplitude of a spring oscillator decreases by 50% after 1 period. What is tgr amplitude after 3 oscillations? What is the time constant assuming that one period is 2seconds?
When the amplidtude is 1/e...1/e=.3676
A=(1/2)^periods=
and then .5^periods=1/e
take ln of each side
periods *ln.5=-1
periods=-1/ln(.5)=1.44
so time constant is1.44*1=2.88seconds
To find the amplitude after 3 oscillations, we need to understand the concept of exponential decay. In an oscillating system such as a spring, the amplitude decreases over time according to a mathematical function related to exponential decay.
The general formula for exponential decay is given by:
A(t) = A0 * e^(-t/τ)
Where:
A(t) is the amplitude after time t
A0 is the initial amplitude
e is the base of the natural logarithm (approximately 2.71828)
t is the time elapsed
τ (tau) is the time constant
In this case, we are given that the amplitude decreases by 50% after 1 period. Since the amplitude decreases, we can assume that A(t) = A0 / 2.
Now, let's substitute the given values into the formula:
A(t) / A0 = e^(-t/τ)
Since A(t) = A0 / 2, we can rewrite the equation as:
(A0 / 2) / A0 = e^(-t/τ)
Simplifying further:
1/2 = e^(-t/τ)
To find the amplitude after 3 oscillations, we need to substitute t = 3 * T (where T is the period) into the equation above. Since the given period is 2 seconds, T = 2 seconds.
t = 3 * 2 = 6 seconds
Now, we can solve for τ:
1/2 = e^(-6/τ)
Next, we isolate τ by taking the natural logarithm (ln) of both sides:
ln(1/2) = -6/τ
Using the properties of logarithms:
ln(1/2) = ln(1) - ln(2) = 0 - ln(2) = -0.6931
Now we can solve for τ:
-0.6931 = -6/τ
Cross multiplying:
-0.6931 * τ = -6
Dividing both sides by -0.6931:
τ = 8.6464
Therefore, the time constant τ is approximately 8.6464 seconds.
Finally, we can calculate the amplitude after 3 oscillations using the equation:
A(t) = A0 * e^(-t/τ)
Substituting A0 / 2 for A(t) and 6 seconds for t:
A(6) = (A0 / 2) * e^(-6/τ)
Substituting the value of τ:
A(6) ≈ (A0 / 2) * e^(-6/8.6464)
Calculating e^(-6/8.6464):
e^(-6/8.6464) ≈ 0.4191
Now, we can plug in the value:
A(6) ≈ (A0 / 2) * 0.4191
Therefore, the amplitude after 3 oscillations is approximately 0.4191 times the initial amplitude.