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.