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i want to ask about the relation highlighted in yellow in the picture the equation of motion of this system can be obtained using the Lagrange formalism, but my question is about this relation i have had some issues in finding it i used fundamental relation of dynamics but i didn't find any result like that. the translation of the text in the image:

in the system of figure 11, the two pulleys we neglect the weight both have the same radius R. find the expressions of the kinetic energy, the potential energy and the equation of motion and the natural frequency of the oscillation. U is the potential energy. T is the kinetic energy.

thanks in advance.

enter image description here

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The part highlighted in yellow is saying that the forces from the two springs are equal. (The force from a spring (Hooks Law) is $F = K x$).

So the force from the first spring $F = k_1 y_1 $ equals that from the second, $F = k_2 y_2 $.

Looking at the blue rope in the figure it feels intuitive that the two forces should be equal. As the mass is increased the top one is pulled further down and the bottom further up. Mechanically the reason the two forces must be equal is in order to set the tension equal throughout the string.

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    $\begingroup$ thanks, I got it. $\endgroup$ – Amine Boudjoghra Sep 19 at 21:37

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