# Mass-spring-dampened system

1. Nov 15, 2012

### phenalor

1. The problem statement, all variables and given/known data
A mass-spring-dampener system is applied a force $mg$ and is immediatly removed, setting the system in motion. The system is constantly applied force $Mg$ and is static at $y=y_0$.
Find a formula for both $A$ and $\phi$

2. Relevant equations

$\ddot{y}+2\delta\dot{y}+w_0^2y=0$
$\frac{2\pi}{w_0}=T_0$
$\delta = \frac{3}{5}w_0$
$F_f=-b\dot{y}$
$Mg=ky_0$

3. The attempt at a solution

from this i find $k$ and $b$. No problem, not part of my question.

when the force is applied, the system 'moves' in y direction and is set in motion, given function:

$y(t)=Ae^{-\delta t}cos\left(w_d t+\phi\right)$
$w_d=\sqrt{w_0^2+\delta^2}$

I'm to find $A$ and $\phi$

my try:

I understand $\dot{y}(0)=0$ gives:
$\dot{y}=-\delta Ae^{-\delta t}cos(w_d t+\phi)-w_d A e^{-\delta t}sin(w_d t + \phi)$
gives:
$\phi=\arctan{\frac{-\delta}{w_d}}$

however i do not find a substitute for $y(0)$. The solution says $y(0)=\frac{m}{M}y_0$, but i don't see the logic in that at all

Sorry if its a bit caotic, this is only part of the assignment. ask and i will provide!

2. Nov 15, 2012

### phenalor

nvm, found the solution