# An interesting coupled oscillators problem (multiple springs and masses)

## Homework Statement:

Waves and oscillations

## Relevant Equations:

2nd Newton's Law.

I need to find the differential equations for each mass. ##y_1## is the equilibrium position, and ##y_2## is the second equilibrium position for each mass.
I was thinking consider the next sistem:

\begin{eqnarray}
k\Delta y-mg&=&m\frac{d^2 y_2}{dt^2}
\\ -2k\Delta y_1 -k\Delta y_2 -2mg &=&m\frac{d^2 y_1}{dt^2}
\end{eqnarray}

But i'm not sure , anybody can help me please??

Last edited:

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haruspex
Homework Helper
Gold Member
\begin{eqnarray}
y_2 (t) &=& k\Delta y-mg=m\frac{d^2 y_2}{dt^2}
\\ y_1(t)&=& -2k\Delta y_1 -k\Delta y_2 -2mg =m\frac{d^2 y_1}{dt^2}
\end{eqnarray}
I don’t understand the ##y_1 (t)=## and ##y_2 (t)=##. Do you mean "For ##y_1##:" etc?
What is "y" in the ##k\Delta y## in ##k\Delta y-mg=m\frac{d^2 y_2}{dt^2}##?

berkeman
Mentor
And for me, the units do not match in these equations (position units versus force units?)

\begin{eqnarray}
\Delta y
\end{eqnarray}
is the variation of each spring, and k is the resort constant,, and i'm sorry, y_1 is the solution, it is my mistake.

berkeman
Mentor
\begin{eqnarray}
\Delta y
\end{eqnarray}
is the variation of each spring, and k is the resort constant,, and i'm sorry, y_1 is the solution, it is my mistake.
No worries. Can you re-write your differential equations so the units are consistent? That will help a lot in moving forward. Thanks.

haruspex
Homework Helper
Gold Member
\begin{eqnarray}
\Delta y
\end{eqnarray}
is the variation of each spring, and k is the resort constant,, and i'm sorry, y_1 is the solution, it is my mistake.
But you wrote ##k\Delta y-mg=m\frac{d^2 y_2}{dt^2}##. So you have ##k\Delta y## on the left but ##y_2## on the right. So presumably you meant ##k\Delta y_2-mg=m\frac{d^2 y_2}{dt^2}##, yes?
But that is incorrect. The length of the middle spring is ##y_1-y_2##, so the variation in its length is the variation in that expression.
Your second equation has several errors. I feel you have jumped straight into writing the differential equations when you would do better to take it in smaller steps. Write variables for the forces in the springs, and write the ##\Sigma F=ma## equation for each mass in terms of those forces and the force of gravity on that mass.

Hi! does this exercise come from a book?