Oscillations notes

Oscillations deals with repeated motion about a mean position. The chapter begins with periodic motion and oscillatory motion, then focuses on simple harmonic motion, where acceleration is directly proportional to displacement and directed towards the mean position. SHM is described using sine or cosine equations, phase, angular frequency, velocity and acceleration. The chapter further connects SHM with restoring force, Hooke’s law, kinetic energy, potential energy and conservation of mechanical energy. Finally, the simple pendulum is studied as an approximate SHM system for small angular displacement. For NEET, this chapter is highly formula-based and commonly tests time period, frequency, spring systems, pendulum variations, energy and graph interpretation.

Periodic motion is any motion that repeats itself after equal intervals of time. The fixed time after which the motion repeats is called time period, and the number of repetitions per second is frequency. Oscillatory motion is a special periodic to-and-fro motion about a mean position. A motion becomes physically important in oscillations when displacement, velocity, acceleration and restoring force change regularly with time. Angular frequency tells how fast the phase of oscillation changes. The important characteristics of oscillations are amplitude, time period, frequency, angular frequency, phase and mean position. NEET questions from this topic usually check definitions, unit conversion and graph interpretation.

Simple Harmonic Motion is a special oscillatory motion in which acceleration is directly proportional to displacement from the mean position and is always directed towards the mean position. Mathematically, a = -ω²x. The negative sign is the signature of restoring nature. SHM can be represented by x = A sin(ωt + ϕ) or x = A cos(ωt + ϕ), where amplitude, angular frequency and phase completely describe the motion. A powerful way to understand SHM is as the projection of uniform circular motion on a diameter. Many systems show approximate SHM, such as a spring-mass system, small oscillations of a pendulum and vibrations of atoms.

In SHM, velocity and acceleration continuously change with position and time. If displacement is x = A sin(ωt + ϕ), then velocity is v = Aω cos(ωt + ϕ), so velocity is maximum at the mean position and zero at extreme positions. Acceleration is a = -ω²x, so it is zero at the mean position and maximum in magnitude at the extremes. Velocity is 90° ahead of displacement in phase, while acceleration is 180° out of phase with displacement. These relations are central for NEET because many questions ask maximum velocity, maximum acceleration, speed at a given displacement and interpretation of v-t and a-t graphs.

Force and energy give the physical cause and behaviour of SHM. A restoring force always acts towards the mean position and tries to bring the particle back. For a spring, Hooke’s law gives F = -kx, which matches the SHM condition because acceleration becomes a = -(k/m)x. During SHM, kinetic energy is maximum at the mean position and zero at extremes, while potential energy is zero at the mean position and maximum at extremes. Total mechanical energy remains constant in ideal SHM and equals 1/2 kA² or 1/2 mω²A². NEET commonly tests energy at a given displacement and conservation of energy in spring systems.

A simple pendulum consists of a small heavy bob suspended by a light, inextensible string from a fixed support. When displaced slightly and released, the bob oscillates about its mean position. For small angular displacement, sinθ ≈ θ, so the restoring torque becomes proportional to angular displacement and the motion is approximately SHM. Its time period is T = 2π√(L/g), where L is the effective length from point of suspension to the centre of the bob. The period depends on length and acceleration due to gravity, but not on mass of the bob. Energy changes between gravitational potential energy at extremes and kinetic energy at the mean position.