The amplitude of an oscillator decreases to 69.0% of its initial value in 15.5s.

a) What is the value of the time constant?

If the time constant is T,

A(t) = A(0) e^(-t/T)

0.69 = e^(-15.5/T)
Take natural logs of both sides.

ln(0.69) = -0.3711 = -15.5/T
T = 41.8 s

To find the value of the time constant, we can use the formula:

A(t) = A(0) * e^(-t/τ)

Where:
A(t) is the amplitude at time t
A(0) is the initial amplitude
τ is the time constant

Given that the amplitude decreases to 69.0% of its initial value, we can rewrite the formula as:

0.69 = e^(-15.5/τ)

To find the value of τ, we need to solve this equation for τ. Taking the natural logarithm (ln) of both sides:

ln(0.69) = ln(e^(-15.5/τ))

Using the property ln(e^x) = x:

ln(0.69) = -15.5/τ

Now, we can solve for τ by rearranging the equation:

τ = -15.5 / ln(0.69)

Calculating this value gives:

τ ≈ -15.5 / ln(0.69) ≈ 25.24 seconds

So, the value of the time constant is approximately 25.24 seconds.

To find the value of the time constant, we need to use the exponential decay formula for the amplitude of an oscillator:

A(t) = A0 * e^(-t / τ)

where A(t) is the amplitude at time t, A0 is the initial amplitude, e is the base of the natural logarithm (approximately 2.71828), t is the time, and τ is the time constant we want to calculate.

Given that the amplitude decreases to 69.0% of its initial value, we can write the equation as:

0.69 * A0 = A0 * e^(-15.5 / τ)

Simplifying the equation, we can cancel out A0 on both sides:

0.69 = e^(-15.5 / τ)

To further simplify the equation, we can take the natural logarithm (ln) of both sides:

ln(0.69) = -15.5 / τ

Now, we can solve for τ by rearranging the equation:

τ = -15.5 / ln(0.69)

Using a calculator, we can calculate the value of the time constant.