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AyushNaman

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For example, consider this problem:

Two blocks of mass 3kg and 6kg are connected by a spring (stiffness=200), with initial velocities of 1m/s towards the left and 2m/s towards the right, respectively. To find the maximum elongation, the most commonly used approach is the "Conservation of Mechanical Energy" method. We calculate the velocity of the center of mass and equate the initial total kinetic energy to the later potential energy plus the later kinetic energy to find the 'x' term, representing the maximum elongation.

Now, the question is, how can one approach such questions using the work-energy theorem? I tried to apply it, but the main issue I encountered was figuring out the expression for the work done by the spring on individual blocks. This confusion arises from the following doubt:

During the derivation of the work done by a spring on a body, we took the spring to be attached to a certain support, such as a wall. Then the spring was connected to a block, displaced by an amount 'x', and then the basic integration followed :$$\int_{0}^{x}kx dx= \frac{kx²}{2}$$

However, the spring here chosen was at rest and the amount of force applied by the spring and the displacement of the block were both related by the 'x' term. What I mean is kx and the x in small displacement dx are the same, or in simple terms, the force on the block by the spring was a function of block's displacement and thus we could integrate.

HOWEVER, in these questions, the spring and blocks are both moving, and the elongation of the spring and the amount of a block's displacement are completely independent. A force kx(x-elongation in the spring) acts on both the blocks, but they don't displace by the same amount,i.e, the force by the spring the blocks now isn't a function on of their displacements, so we can't integrate and just write the regular {kx²/2} expression. Also, we can't simply integrate kx*ds without knowing the actual displacements of the blocks. Some might say it's 1/2k(x_1)^2 and 1/2k(x_2)^2 for individual blocks. But as I already described, the displacement and the elongation have no relation, so it's not that straightforward.

Also if the mechanical energy conservation approach is used, it is directly assumed that the potential energy in the spring is 1/2kx²(x being the total elongation in the spring). The proof of it is, in fact, based on the assumption that the spring is fixed, and the force acting on the spring is a function of its endpoint displacement. In this case, if one were to prove it from scratch, they cannot do so since both ends are moving with separate velocities, resulting in different displacements but under the same force, k * (elongation). So we would need to actually find out the work done by this conservative spring force.

This issue has left me confused, and despite asking my teacher and many others, no one could interpret my doubt. All the solutions provided to such questions are only using the conservation of mechanical energy. I would be utterly grateful if you could look into this and provide clarification.