In the figure below, a damped simple harmonic oscillator has mass m = 220 g, k = 95 N/m, and b = 70 g/s. Assume all other components have negligible mass. What is the ratio of the amplitude of the damped oscillations to the initial amplitude at the end of 20 cycles (Adamped / Ainitial)?
I will be happy to critique your thinking.
To find the ratio of the amplitude of the damped oscillations to the initial amplitude at the end of 20 cycles, we need to calculate the amplitude of the damped oscillations and the initial amplitude first.
Amplitude of damped oscillations:
The equation for the amplitude of a damped simple harmonic oscillator is given by:
A_damped = A_initial * exp(-bt/(2m))
where A_damped is the amplitude of the damped oscillations, A_initial is the initial amplitude, b is the damping coefficient, t is the time, and m is the mass.
We are given:
m = 220 g (convert to kg: m = 0.22 kg)
b = 70 g/s (convert to kg/s: b = 0.07 kg/s)
A_initial is not given, so we need to find it.
Initial amplitude:
The initial amplitude can be found using the following formula for simple harmonic motion:
A_initial = m * ω^2 / k
where ω is the angular frequency, which can be calculated using the formula: ω = sqrt(k/m)
k = 95 N/m (convert g to kg: k = 95 kg/s^2)
Calculating the angular frequency:
ω = sqrt(k/m) = sqrt(95/0.22) ≈ 13.12 rad/s
Substituting the values into the formula for A_initial:
A_initial = (0.22 kg) * (13.12 rad/s)^2 / (95 N/m)
= 0.22 * (13.12)^2 / 95 ≈ 0.403 m
Now that we have the values for A_damped and A_initial, we can find the ratio:
Ratio (Adamped / Ainitial) = A_damped / A_initial
= A_initial * exp(-bt/(2m)) / A_initial
= exp(-bt/(2m))
Substituting the given values into the formula:
Ratio = exp(-0.07 kg/s * 20 cycles / (2 * 0.22 kg))
= exp(-0.7)
Using a scientific calculator, we find that the value of e^-0.7 is approximately 0.496.
Therefore, the ratio of the amplitude of the damped oscillations to the initial amplitude at the end of 20 cycles is approximately 0.496.