An interesting coupled oscillators problem (multiple springs and masses)

  • #1

Homework Statement:

Waves and oscillations

Relevant Equations:

2nd Newton's Law.
Sin título.png

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:

Answers and Replies

  • #2
haruspex
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\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}##?
 
  • #3
berkeman
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And for me, the units do not match in these equations (position units versus force units?)

1593297692082.png
 
  • #4
\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.
 
  • #5
berkeman
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58,024
8,081
\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. :smile:
 
  • #6
haruspex
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\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.
 
  • #7
Hi! does this exercise come from a book?
 

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