Ratio of Damped to Initial Oscillation Amplitudes - 20 Cycles

[bclark23]

In the figure below, a damped simple harmonic oscillator has mass m = 300 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 know I need to find the period (T), which is 2πsqrt(m/k).
T=2πsqrt[(.0kg)/(95nN/m)]=.353 s

Also, there are 20 cycles, so the final time would be (20 cycles)(.353s)=7.062s

The formula for damping (Adamped?) is x(t)=x_me^-bt/2mcos(wt+rho)
The formula for oscillation (Ainital?) is x(t)=x_mcos(wt+rho)

I'm pretty sure I need to use these two equations, and put the answers in a ratio, but I'm not sure how to go about doing that.

 

[haruspex]

The formula for damping (Adamped?) is x(t)=x_me^-bt/2mcos(wt+rho)

Don't you just substitute for b, t and m in e^-bt/2m to find the attenuation?

 

[bclark23]

Yes, that was exactly it! Thank you