Q.A block slides along a rough horizontal surface and gradually comes to rest. Using the work-energy theorem, explain why its kinetic energy decreases even though no external agent is removing energy from the block-surface system as a whole, and state where the lost mechanical energy actually goes.

By the work-energy theorem, the change in the block's kinetic energy always equals the net work done on it: $\Delta K = W_{\text{net}}$. As the block slides, the only horizontal force acting on it is kinetic friction, which by its very nature always opposes the direction of relative sliding -- so friction here acts directly opposite to the block's velocity at every instant. This makes the work done by friction on the block negative ($\theta=180^\circ$ in $W=Fd\cos\theta$), and so $\Delta K = W_{\text{net}} < 0$: the block's kinetic energy must decrease, exactly as observed, and it continues decreasing until the block comes to rest (at which point friction, having nothing left to oppose, does no further work).

This does not violate the more general law of conservation of energy (energy overall is never created or destroyed) -- it only means that mechanical energy specifically ($K+U$, and here $U$ does not even change, since the motion is purely horizontal) is not conserved for the block, because friction is a non-conservative force. The kinetic energy the block loses is not annihilated; it is converted, at the microscopic level of the sliding contact surfaces, into an increase in internal (thermal) energy of the block and the floor -- felt macroscopically as the surfaces becoming slightly warmer -- along with a very small amount that may escape as sound. This is exactly the general lesson of §5.6 and §5.8: whenev …

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