OSCILLATIONS AND WAVES OBJECTIVE 5

Force Derivation (Option C)

The force exerted by a particle moving with speed \(v\) in a length \(L\) is \(F = mv^2/L\). With the piston displaced by \(x\):

$$ F_{net} = \frac{mv^2}{L+x} – \frac{mv^2}{L-x} \approx -\frac{2mv^2}{L^2} x $$

Since \(F \propto -x\), this represents Simple Harmonic Motion.

Alternative Perspective (Option B)

While the ideal elastic model predicts eternal SHM, Option (B) is also considered correct in physical contexts where the “heavy” piston (\(M \gg m\)) interacts with “light” particles. Over a very long time, statistical fluctuations or non-ideal energy transfer (damping/entropy maximization) can cause the macroscopic piston to settle at the equilibrium position (center) and stop.