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Coupled Flarmonic Oscillators X2 : 0 ) Write down the 2nd law for each of the masses. Use coordinates x, aud x, for m and M, and i,, ×2 and a, az. timt: the force from the ferce from spring 2 &on m | on m'only depends but on X,, spring depand s on (x2-x,)

Coupled Flarmonic Oscillators X2 : 0 ) Write down the 2nd law for each of the masses. Use coordinates x, aud x, for m and M, and i,, ×2 and a, az. timt: the force from the ferce from spring 2 &on m | on m'only depends but on X,, spring depand s on (x2-x,)

Classical Dynamics of Particles and Systems
Classical Dynamics of Particles and Systems
5th Edition
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Stephen T. Thornton, Jerry B. Marion
Chapter8: Central-force Motion
Section: Chapter Questions
Problem 8.7P
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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).
Transcribed Image Text: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).
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Given, masses m and M. Where, M>m.

And, x1 and x2  are the respective displacements of m and M from its equilibrium position.

We deduce, x2>x1 

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

For mass m:

md2x1dt2=-k1x1-k2x2-x1md2x1dt2=-k1+k2x1+k2x2

For mass M:

Md2x2dt2=-k3x2-k2x2-x1Md2x2dt2=-k2+k3x1+k2x1

 

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