CoupledHarmonicOscillatorsX2 :0eeeeeLeeee NM) Write down the 2nd law for each of the masses.Use coordinates x, aud x, for m and M, and i,, *2 and a, a2.Hirnt: the force from spring I on monly dependson x, , butthe ferce from spring 2 on mdepand son (x2-x,)two differantial2) To simplify, let k, -k,k and m-M. Rewriteegg equatious from i) right above each other.your3) Detine two new variables, X (Greek"chi") - x,+X2 and Ax = X,-x 2two di ferential equations to produceThen add and subtractyourvery simple (harmanic) ones.4) Write down the solutiou to bothe equations usingWask+2k2 ond evefficients A, B, C, and Dtwo new ,butw. :un5) Now assune k-10k2 (this means that the two masses are"weakly coupled"). Also assume X,(0) = -10cm, x,(0)=0, xz (0)= o,x,(0)=0. Solve for A,B, C, D, and solve for x (t).

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College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

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Coupled
Harmonic
Oscillators
X2 :0
eeeee
Leeee NM
) Write down the 2nd law for each of the masses.
Use coordinates x, aud x, for m and M, and i,, *2 and a, a2.
Hirnt: the force from spring I on monly depends
on x, , but
the ferce from spring 2 on m
depand s
on (x2-x,)
two differantial
2) To simplify, let k, -k,k and m-M. Rewrite
egg equatious from i) right above each other.
your
3) Detine two new variables, X (Greek"chi") - x,+X2 and Ax = X,-x 2
two di ferential equations to produce
Then add and subtract
your
very simple (harmanic) ones.
4) Write down the solutiou to bothe equations using
Wask+2k2 ond evefficients A, B, C, and D
two new ,
but
w. :
un
5) Now assune k-10k2 (this means that the two masses are
"weakly coupled"). Also assume X,(0) = -10cm, x,(0)=0, xz (0)= o,
x,(0)=0. Solve for A,B, C, D, and solve for x (t).

Expert Solution

Step 1

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Step 2 (1)

Given, masses m and M. Where, M>m.

And, $x_{1} and x_{2}$ are the respective displacements of m and M from its equilibrium position.

We deduce, $x_{2} > x_{1}$

So, writing Newton's second law of motion for both the masses.

For mass m:

$\begin{array}{rcl} m \frac{d^{2} x_{1}}{d t^{2}} & = & - k_{1} x_{1} - k_{2} (x_{2} - x_{1}) \\ m \frac{d^{2} x_{1}}{d t^{2}} & = & - (k_{1} + k_{2}) x_{1} + k_{2} x_{2} \end{array}$

For mass M:

$\begin{array}{rcl} M \frac{d^{2} x_{2}}{d t^{2}} & = & - k_{3} x_{2} - k_{2} (x_{2} - x_{1}) \\ M \frac{d^{2} x_{2}}{d t^{2}} & = & - (k_{2} + k_{3}) x_{1} + k_{2} x_{1} \end{array}$

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