Damped Oscillations

Damped oscillation, treated in NCERT Class 11, Chapter 14 (Oscillations), arises when a resistive force $F=-bv$ opposes motion in addition to the restoring force. The resulting displacement is $x(t)=A{e}^{-bt/2m}\cos ({\omega }^{\prime }t+\varphi )$, in which the amplitude term $A{e}^{-bt/2m}$ shrinks exponentially with time. A car's shock absorber is exactly this: it damps out the oscillation of the suspension so the ride settles. Hence the amplitude falls off exponentially, most rapidly at first and then more gradually. The option that amplitude stays constant while frequency falls is wrong; although the frequency does shift slightly to ${\omega }^{\prime }$, the amplitude clearly decreases. The option that amplitude increases toward resonance describes forced oscillations, not free damping. The option of a linear drop is wrong because the decay is governed by an exponential factor, not a straight-line decrease. It is instructive to see why the decay is exponential rather than linear: the resistive force is proportional to velocity, so the rate of energy loss depends on how fast the object is currently moving, and a loss rate proportional to the amount present always yields exponential behaviour. Since the mechanical energy scales as the square of amplitude, the energy itself decays as $e^{-bt/m}$, twice as fast in the exponent as the amplitude. A consistency check: the exponent $-bt/2m$ is dimensionless as required, because $b$ has units of kg/s so $bt/m$ is unitless, and energy is continually lost to friction, so a monotonic exponential decrease is exactly what is physically expected.