In a damped oscillator with m = 248 g, k = 80.6 N/m, and b = 67.4 g/s, what is the ratio of the amplitude of the damped oscillations to the initial amplitude at the end of 20 cycles?
To find the ratio of the amplitude of the damped oscillations to the initial amplitude at the end of 20 cycles, we need to use the formula for the amplitude of damped oscillations given by:
A(t) = A(0) * e^(-bt/2m)
Where:
A(t) is the amplitude at time t
A(0) is the initial amplitude
b is the damping coefficient
m is the mass of the oscillator
t is the time
In this case, we are interested in the amplitude at the end of 20 cycles, so we can substitute t with the time it takes for 20 cycles to complete.
First, we need to find the time it takes for one cycle to complete. The formula for the time period of an oscillator is given by:
T = 2π * √(m/k)
Where T is the time period, m is the mass, and k is the spring constant.
Substituting the given values, we have:
T = 2π * √(0.248 kg / 80.6 N/m)
Now we can find the time it takes for 20 cycles:
T20 = 20 * T
Next, we can substitute the values for b, m, and t into the formula for amplitude:
A(t) = A(0) * e^(-bt/2m)
Given:
m = 248 g = 0.248 kg
k = 80.6 N/m
b = 67.4 g/s = 0.0674 kg/s
t = T20
Finally, we can calculate the ratio of the amplitude of the damped oscillations to the initial amplitude:
Amplitude Ratio = A(T20) / A(0)