How can I uncouple this system when m1 ≠ m2? ....

[juan carlos trujillo]

this is the exercise :

i've got by modes:

https://scontent.fbog2-1.fna.fbcdn.net/v/t35.0-12/15311417_1271935189529554_1902854011_o.jpg?oh=09710fddf04ae58a465238d8f1786189&oe=58424904

but i can't uncouple it.

this is supposed to in the form x = -w² *x i would like if you can help me to solve this problem.

 

[mfb]

Did you try the general linear approach $u=x_1 + c x_2$ with some constant c? Ideally, two values for c give (different) uncoupled equations.

 

[juan carlos trujillo]

No, I haven't. Because i really don't know that eqauation.

what i have tried is when applying the sum and substraction (x1+x2) and (x1-x2) as well as the tecnique C1/C2. But none of them gives the frecuency i need due to the masses that are differents and at the end i ended up with nothing because of I got to a point where i couldn't simplify anything.

i've heard a possible solution for this problem could be done by solving through Fourier transform.But i don't know how to apply it .

 

[mfb]

Physics and mathematics are not about memorizing equations. It is about figuring out how to solve a problem.

The sum and differences don't allow to decouple the system, as you noted already.

Fourier transformation should work as well. Just transform the equations.

Guessing a solution and then figuring out the constants is another option: for each mass it will be the sum of two oscillations.

 

[haruspex]

Because i really don't know that eqauation.

mfb's suggsted method is quite straightforward. The idea is that there is some point between the masses, maybe not the common mass centre but some other weighted combination of the x values, which satisfies a simpler equation.
If you define u as in post #2 you can get an equation for $\ddot u$ in terms of the x's and c. If you simply assume that expression can be rewritten as some constant times u you can get an equation for c interms of k and the two masses.
By the way, the algebra will be simpler if you first replace k/m₁ by k₁, similarly k₂.

 

[ehild]

this is the exercise :

i've got by modes:

but i can't uncouple it.

this is supposed to in the form x = -w² *xi would like if you can help me to solve this problem.

The usual procedure to solve such problems is to assume the solution in the form x₁=ae^iωt and x₂=be^iωt. Substituting into the system of differential equations, you get an algebraic equation for the angular frequency ω. There will be two possible frequencies, and according to them, two possible "normal modes" -two dimensional vectors with components x₁ and x₂.