Chapter 13: Oscillations | Summary

Simple harmonic motion (SHM) is the simplest form of oscillatory motion, in which the displacement of a particle from its equilibrium position varies sinusoidally with time: x(t) = A cos(ωt + φ). Here, A is the amplitude (maximum displacement), ω is the angular frequency, (ωt + φ) is the time-dependent phase, and φ is the phase constant. The period is T = 2π/ω and the angular frequency ω = 2πν.

Two SHMs can share the same angular frequency and phase but differ in amplitude, or share the same amplitude and angular frequency but differ in phase constant. The phase constant φ is determined by the initial conditions of displacement and velocity at t = 0. SHM is not just any periodic motion but specifically one where displacement is a sinusoidal function of time.

SHM can be understood as the projection of uniform circular motion onto a diameter of the reference circle. If a particle P moves uniformly on a circle of radius A with angular speed ω, its projection P′ on the x-axis executes SHM: x(t) = A cos(ωt + φ). Similarly, the projection on the y-axis gives y(t) = A sin(ωt + φ), which is also SHM but with a phase difference of π/2.

Despite this geometric connection, the force on a particle in linear SHM is fundamentally different from the centripetal force that maintains uniform circular motion. The reference circle is a mathematical tool to visualise and analyse SHM, not a description of the actual physical path of the oscillating particle.

The instantaneous velocity of a particle in SHM is v(t) = −ωA sin(ωt + φ), and the instantaneous acceleration is a(t) = −ω²A cos(ωt + φ) = −ω²x(t). The velocity amplitude is ωA and the acceleration amplitude is ω²A. Both velocity and acceleration are sinusoidal functions of time with the same period T as displacement.

The velocity leads the displacement by a phase of π/2, and the acceleration leads the displacement by a phase of π (i.e., acceleration is always directed opposite to displacement). At the mean position (x = 0), speed is maximum (ωA) and acceleration is zero. At the extreme positions (x = ±A), speed is zero and the magnitude of acceleration is maximum (ω²A).

Using Newton's second law, the restoring force on a particle of mass m in SHM is F(t) = ma = −mω²x(t) = −kx(t), where k = mω² is the force constant. This force is always directed toward the mean position, which is why it is called the restoring force. The angular frequency is related to the force constant by ω = √(k/m).

A particle oscillating under a force linearly proportional to displacement is called a linear harmonic oscillator. In real systems, small additional terms proportional to x², x³, etc., may be present, giving rise to non-linear oscillators. SHM can be defined either through its displacement equation x(t) = A cos(ωt + φ) or equivalently through the force law F = −kx.

In SHM, kinetic energy is K = ½mω²A² sin²(ωt + φ) = ½kA² sin²(ωt + φ), and potential energy is U = ½kA² cos²(ωt + φ). Both K and U vary periodically with period T/2. Kinetic energy is maximum (= ½kA²) at the mean position and zero at the extremes; potential energy is zero at the mean position and maximum (= ½kA²) at the extremes.

The total mechanical energy E = K + U = ½kA² is constant and independent of time, as expected for a conservative force system. Both kinetic and potential energies are always non-negative. At any intermediate position, energy is shared between kinetic and potential forms, with one increasing at the expense of the other.

A simple pendulum consists of a small bob of mass m tied to a massless inextensible string of length L, fixed at the top. When displaced by a small angle θ from the vertical, the restoring torque is τ = −mgL sin θ. For small angles, sin θ ≈ θ (in radians), and the equation of motion becomes α = −(mgL/I)θ, which is mathematically identical to the equation for SHM.

Since the string is massless, the moment of inertia I = mL², and the time period simplifies to T = 2π√(L/g). This period is independent of the mass of the bob and the amplitude (for small oscillations). A seconds pendulum, which ticks every second (T = 2 s), has a length of approximately 1 m under standard gravity.

Chapter 13 establishes oscillatory motion as a foundational topic in physics, built around the central concept of simple harmonic motion. SHM is uniquely defined by a restoring force proportional to displacement, leading to sinusoidal variation of displacement, velocity, and acceleration with time. The total mechanical energy in SHM remains constant, continuously exchanged between kinetic and potential forms. The simple pendulum is a practical realisation of SHM for small amplitudes, with a period that depends only on its length and the local gravitational field.

From a board-exam perspective, this chapter is highly important. Students must be able to write and interpret the SHM displacement equation, derive expressions for velocity, acceleration, and energy, apply the force law, and solve numericals on the simple pendulum. The relationship between SHM and uniform circular motion, the phase relationships among x, v, and a, and the constancy of total energy are frequently tested concepts in CBSE examinations.