Board exam focus — Oscillations (CBSE & HBSE)

Oscillations introduces periodic and oscillatory motion, with simple harmonic motion (SHM) as the central idea. This chapter develops the kinematics of SHM (displacement, velocity, acceleration, phase), the energy stored in an oscillating system, the linear restoring force law, and standard systems such as the simple pendulum and the loaded spring. It closes with free, damped, forced oscillations and the phenomenon of resonance — concepts that recur in Waves, AC circuits and modern physics.

Periodic Motion and Simple Harmonic Motion

Periodic and Oscillatory Motion

A motion that repeats itself after equal intervals of time is called periodic motion. The smallest such interval is the time period T. If the body moves to and fro about a fixed mean (equilibrium) position, the periodic motion is called oscillatory (or vibratory) motion.

• Every oscillatory motion is periodic, but every periodic motion need not be oscillatory (e.g. uniform circular motion is periodic but not oscillatory).
• Examples of oscillatory motion: a swinging pendulum, a vibrating tuning fork, a mass on a spring.

Simple Harmonic Motion (SHM)

SHM is the simplest form of oscillatory motion in which the restoring force is directly proportional to the displacement from the mean position and is always directed towards the mean position.

The displacement of a particle executing SHM is:

$$x(t) = A\,\sin(\omega t + \phi)$$

Velocity and Acceleration

Differentiating displacement:

• Velocity: v = dx/dt = Aω cos(ωt + φ) = ω√(A² − x²)
• Acceleration: a = dv/dt = −Aω² sin(ωt + φ) = −ω²x

The defining relation of SHM is therefore:

$$a = -\omega^2 x$$

Acceleration is maximum (Aω²) at the extreme positions and zero at the mean position, while velocity is maximum (Aω) at the mean position and zero at the extremes.

Trap (CBSE/HBSE): Acceleration in SHM is NOT constant. It is proportional to displacement and oppositely directed. The negative sign indicates direction towards the mean position.

SHM as Projection of Uniform Circular Motion

SHM can be visualised as the projection of a particle moving uniformly on a circle (the reference circle) of radius A onto a diameter. This geometric picture explains why sine/cosine functions describe SHM and why ω is called the angular frequency.

Energy in SHM and the Force Law

Restoring Force and Force Law

For a particle of mass m in SHM, the acceleration is a = −ω²x. By Newton's second law the force is:

$$F = ma = -m\omega^2 x = -k x$$

where k = mω² is the force constant (restoring force per unit displacement). Hence:

$$\omega = \sqrt{\frac{k}{m}}, \qquad T = 2\pi\sqrt{\frac{m}{k}}$$

The restoring force is linear in displacement — this linearity is the hallmark of SHM.

Kinetic Energy (KE)

$$KE = \tfrac{1}{2}mv^2 = \tfrac{1}{2}m\omega^2(A^2 - x^2)$$

• KE is maximum at the mean position (x = 0): KE_max = ½mω²A².
• KE is zero at the extremes (x = ±A).

Potential Energy (PE)

$$PE = \tfrac{1}{2}kx^2 = \tfrac{1}{2}m\omega^2 x^2$$

• PE is zero at the mean position and maximum at the extremes: PE_max = ½mω²A².

Total Mechanical Energy

$$E = KE + PE = \tfrac{1}{2}m\omega^2 A^2 = \text{constant}$$

Trap: Both KE and PE in SHM vary with double the frequency of the displacement (period T/2), yet their sum (total energy) stays constant and is proportional to A² and to ω².

The energy graph versus displacement shows PE as an upward parabola, KE as an inverted parabola, and total energy as a horizontal line.

Pendulum, Spring, and Damped & Forced Oscillations

Simple Pendulum

A simple pendulum consists of a point mass (bob) suspended by a light inextensible string of length L. For small angular displacements (θ < ~10°), the restoring torque gives SHM with:

$$T = 2\pi\sqrt{\frac{L}{g}}$$

• The period is independent of mass and amplitude (for small angles).
• It depends on length L and acceleration due to gravity g.

Trap: The small-angle approximation sin θ ≈ θ is essential; for large amplitudes the motion is periodic but not simple harmonic.

Oscillations of a Spring

For a mass m attached to a spring of force constant k:

$$T = 2\pi\sqrt{\frac{m}{k}}$$

The period is independent of g — a spring-mass oscillator works the same in a satellite (weightlessness). Springs in series give a smaller effective k (1/k = 1/k₁ + 1/k₂), while springs in parallel give k = k₁ + k₂.

Free, Damped and Forced Oscillations

Damped SHM: With a damping force F = −bv, the displacement is:

$$x(t) = A\,e^{-bt/2m}\cos(\omega' t + \phi), \quad \omega' = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}$$

The amplitude decays as e^(−bt/2m) and mechanical energy decays as e^(−bt/m).

Resonance: In forced oscillation, the amplitude becomes maximum when the driving frequency equals the natural frequency (ω_d = ω₀). Smaller damping gives a sharper, taller resonance peak.

Application: Resonance explains why soldiers break step on bridges and how radios tune to a station — and why the Tacoma Narrows bridge collapsed.

Frequently asked questions

Are these Oscillations notes free?

Yes — the Oscillations notes for Physics (Class 11) on Siksha Sarovar are completely free to read, with no account required.

Do these notes follow CBSE and HBSE?

Yes. The Oscillations notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.

What does the Oscillations chapter cover?

Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Oscillations.