Oscillations: Damped Block homework

AI Thread Summary
The discussion revolves around a physics homework problem involving a damped oscillating mass attached to a spring and submerged in water. The key questions are about calculating the time it takes for the amplitude to reduce to one-third of its initial value and determining the number of oscillations during that time. Participants clarify the significance of the water depth and the initial conditions of the mass's displacement and velocity. The relevant equations for amplitude decay and angular frequency are provided, and a solution for the time required is identified as approximately 14.4 seconds. The conversation also touches on whether the period of oscillation changes due to damping, indicating a need for further exploration of frequency adjustments in damped systems.
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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?

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

Meaning at t=0, x=0.8 cm and v=0
 
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?
 
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.
 
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!
 
Is the frequency changed during the damping? :)
 
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