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Can someone help me with this?

Imagine we have a system composed of three spheres 1, 2 and 3.

1 is connected to a wall on its left side with a spring and to sphere 2 with another spring, the sphere 2 is connected with 3 with another one and 3 is connected to a wall on its right side also with a spring. All springs have constant $k$.

1st Doubt. Now, if we have $m_{1}->∞$ what changes in this system? What I really what to know is if the sphere 1 moves at all.

Here's what I thought: It can't move, because if it does the energy of the system would be infinite. I drew this conclusion from:

$$E=\sum_{i=1}^3\frac{1}{2}m_{i}(\frac{dx_{i}}{dt})^2+\frac{1}{2}kx_{i}^2$$

and since $m_{1}$ is considered infinite, the energy would have to be infinite if sphere 1 moved. Therefore, for the energy NOT to be infinite I concluded that sphere 1 couldn't move.

2nd Doubt. If the previous thinking was correct we can simplify the system to just the spheres 2 and 3 now can't we? And consider the sphere 1 to be like... a wall, or something. Am I right?

If I wrote this too hard to understand please let me know. I'll try to explain better. Thanks very much.

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One can always transform to the rest frame of reference of an arbitrarily large inertial mass and eliminate your first doubt; kinetic energy is frame dependent.

For you second doubt, note that as the inertial mass of sphere 1 becomes much larger than the others, assuming no other changes, the acceleration of sphere 1 becomes much smaller than the other spheres and is thus insignificant. In the limit of "infinite" inertial mass, the acceleration of sphere 1 goes to zero.

Thus, as you correctly conclude, only the dynamics of spheres 2 and 3 need be considered.

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