To calculate the ratio of the amplitude of the damped oscillations to the initial amplitude at the end of 22 cycles, we need to use the equation for the amplitude of a damped oscillator. The equation is given by:
A(t) = A₀ * exp(-βt) * cos(ω'dt + ϕ)
Where: A(t) is the amplitude at time t, A₀ is the initial amplitude, β is the damping coefficient (in kg/s), t is the time, ω'd is the damped angular frequency, and ϕ is the phase constant.
The damped angular frequency is given by:
ω'd = sqrt(ω₀² - β²/4m²)
Where: ω₀ is the natural angular frequency (sqrt(k/m)), β is the damping coefficient (in kg/s), and m is the mass of the oscillator (in kg).
First, let's convert the given values to the appropriate units: Mass (m) = 310 g = 0.31 kg Spring constant (k) = 58 N/m Damping coefficient (β) = 66 g/s = 0.066 kg/s
We can calculate the natural angular frequency (ω₀) using the mass and spring constant:
ω₀ = sqrt(k/m)
Next, we can calculate the damped angular frequency (ω'd) using the damping coefficient, mass, and natural angular frequency:
ω'd = sqrt(ω₀² - β²/4m²)
Once we have the damped angular frequency, we can calculate the amplitude at the end of 22 cycles (A(22T)), where T is the time period of one cycle. Since the damping affects the amplitude, we need to evaluate the amplitude at the end of each cycle to consider the damping effect. The ratio of A(22T) to A₀ will give us the desired result.
Please note that the calculation involves trigonometric functions, and the phase constant (ϕ) is not provided in the given information. Without the phase constant, it's not possible to determine the precise values for each cycle. However, you can follow the steps outlined above to calculate the ratio if you have the complete information, including the phase constant.