Oscillations

Almost everything in nature that is stable repeats: a pendulum, a plucked string, an atom in a crystal, the current in a radio circuit, the pressure in your ear as a sound arrives. This chapter is about the simplest and by far the most important kind of repetition. It begins by separating periodic motion from oscillatory motion and fixing the four quantities that describe both — period $T$, frequency $\nu = \frac{1}{T}$, displacement measured from the mean position, and amplitude — together with the angular frequency $\omega = 2\pi\nu$, whose confusion with $\nu$ costs more marks in this chapter than any other single slip. Then simple harmonic motion, defined by $$x(t) = A\cos(\omega t + \phi)$$ and viewed a second way as the shadow of uniform circular motion on a diameter, the reference circle that turns hard timing questions into easy geometry. Differentiating twice gives $v = -\omega A\sin(\omega t+\phi)$ and $a = -\omega^2 x$ — the last being the real definition of SHM, since any motion whose acceleration is proportional to displacement and directed back towards the centre is simple harmonic. Multiplying by the mass turns that into the force law $F = -kx$, so every linear restoring force gives $\omega = \sqrt{\frac{k}{m}}$ and $T = 2\pi\sqrt{\frac{m}{k}}$ — a period independent of amplitude, and independent of $g$ even for a vertical spring, provided displacement is measured from the new equilibrium. Spring combinations follow, parallel adding stiffness and series adding compliance. The energy is $E = \frac{1}{2}kA^2$, constant in total but sloshing between kinetic and potential at twice the frequency of the motion, in a parabolic well that is the reason harmonic motion appears everywhere in physics. The simple pendulum gives $T = 2\pi\sqrt{\frac{L}{g}}$ under the small-angle approximation, with the second's pendulum and the lift, car and liquid variants worked through. Finally the two topics that describe real oscillators rather than ideal ones: damped oscillations, where a resistive force $-bv$ makes the amplitude decay as $e^{-bt/2m}$ and the energy twice as fast, and forced oscillations and resonance, where a driven system responds most violently when the driving frequency matches its natural one — the effect behind a tuned radio, a pumped swing, a shattered wine glass and a collapsed bridge. Topics the rationalised syllabus trimmed — damped oscillations, forced oscillations and resonance, the series spring combination and equivalent-stiffness rules, and the second's pendulum with the effective-gravity variants — are restored in full, because Boards, JEE Main and NEET ask about them every year. Dedicated JEE and NEET Corners follow, each with a full-length exam-pattern practice drill.