Oscillations: Damped Block homework

[SuperCass]

Homework Statement

The drawing to the left shows a mass m= 1.9 kg hanging from a spring with spring constant k = 6 N/m. The mass is also attached to a paddle which is emersed in a tank of water with a total depth of 34 cm. When the mass oscillates, the paddle acts as a damping force given by -b(dx/dt) where b= 290 g/sec. Suppose the mass is pulled down a distance 0.8 cm and released.

a) What is the time required for the amplitude of the resulting oscillations to fall to one third of its initial value?

b) How many oscillations are made by the block in this time?

Homework Equations

x(t) = (Xm)(e^(-bt/2m))cos($$\omega$$'t + $$\phi$$)
$$\omega$$' = $$\sqrt{(k/m)-((b^2)/(4m^2))}$$

The Attempt at a Solution

I'm not sure where to start. Is the water depth significant? What should $$\phi$$ be?

Thanks so much for your help!

 

[rock.freak667]

Suppose the mass is pulled down a distance 0.8 cm and released.

Meaning at t=0, x=0.8 cm and v=0

 

[SuperCass]

So once I plug that into the equation, I get that the original amplitude is .8 cm. So the amplitude I'm finding is one third of that. But if I try to solve for t, I still have the unknown variable x(t)! What should I do?

 

[hikaru1221]

a - Amplitude: $$A(t)=A(0)e^{-bt/2m}$$. You have b, you have m, you have the ratio of the later amplitude and the initial amplitude, can you get t? So do you have to know x and A(0)?

b - Getting the period T, you should get the numbers of oscillations it makes in t.

 

[SuperCass]

Got part a!
Thanks, I didn't know that equation!
so t=14.3956s.

For part b, how would I solve for those oscillations?
I think I'm supposed to find the period and divide the time found in a by that, but does the period change if it's damped? Or am I just wrong here?

Thanks again!

 

[hikaru1221]

Is the frequency changed during the damping? :)