Periodic Motion

SYLLABUS

Equation of simple harmonic motion (SHM), Energy in SHM, Application of SHM: vertical oscillation of mass suspended from coiled spring, Angular SHM, simple pendulum, Oscillatory motion: Damped oscillation, Forced oscillation and resonance.

What is This Chapter About?

Look around you and you will find repetition everywhere in nature. The swing of a pendulum, the back-and-forth vibration of a guitar string, the rise and fall of ocean tides, the rhythmic beating of the human heart — all of these are examples of periodic motion, motion that repeats itself at regular intervals of time.

Chapter 2 of NEB Grade 12 Physics explores this fascinating class of motion in depth, focusing especially on Simple Harmonic Motion (SHM) — the most fundamental and mathematically elegant form of oscillation. With 6 teaching hours and consistent presence in NEB board examinations, this chapter demands both conceptual clarity and strong problem-solving skills.

Key Concepts You Will Learn

Simple Harmonic Motion (SHM)

A body is said to execute simple harmonic motion when its acceleration is always directed toward a fixed equilibrium point and is directly proportional to its displacement from that point. Mathematically: [latex]a = -\omega^2 x[/latex], where the negative sign indicates the restoring nature of acceleration.

The displacement in SHM is described by [latex]x = A\sin(\omega t + \phi)[/latex], where [latex]A[/latex] is the amplitude, [latex]\omega[/latex] is the angular frequency, and [latex]\phi[/latex] is the initial phase. Understanding this equation — what each term means and how changing one affects the motion — is the heart of this chapter.

Energy in SHM

A particle in SHM continuously exchanges energy between kinetic and potential forms. At the equilibrium position, kinetic energy is maximum and potential energy is zero. At the extreme positions (amplitude), kinetic energy is zero and potential energy is maximum. The total mechanical energy [latex]E = \frac{1}{2}m\omega^2 A^2[/latex] remains constant throughout the motion, beautifully illustrating conservation of energy.

Mass-Spring System

One of the most important applications of SHM is a mass suspended from a coiled spring. When displaced and released, the mass oscillates with a period [latex]T = 2\pi\sqrt{\frac{m}{k}}[/latex], where [latex]m[/latex] is the mass and [latex]k[/latex] is the spring constant. This result is derived from first principles in this chapter and forms the basis of many numerical problems.

Angular SHM and the Simple Pendulum

A simple pendulum — a small bob suspended by a light inextensible string — executes angular SHM for small angles of displacement. Its period is [latex]T = 2\pi\sqrt{\frac{l}{g}}[/latex], where [latex]l[/latex] is the length of the pendulum and [latex]g[/latex] is gravitational acceleration. This elegant result is independent of mass and amplitude (for small angles), which is why pendulums were historically used as timekeeping devices.

Damped Oscillation

In the real world, no oscillation continues forever. Air resistance and friction gradually remove energy from the system, causing the amplitude to decrease over time. This is damped oscillation. The chapter introduces the concept qualitatively, distinguishing between underdamped, critically damped, and overdamped cases.

Forced Oscillation and Resonance

When an external periodic force is applied to an oscillating system, the result is forced oscillation. When the frequency of the external force matches the system’s natural frequency, a dramatic phenomenon called resonance occurs — the amplitude of oscillation becomes very large. Resonance has profound real-world consequences, from the design of bridges and buildings to the tuning of musical instruments and radio receivers.

Why This Chapter Matters for NEB Exams

Periodic Motion is one of the most reliably examined chapters in NEB Grade 12 Physics. Derivations of the period of a simple pendulum and the mass-spring system appear almost every year. Energy in SHM — showing that total energy is constant — is another frequent long-answer question. Numerical problems involving amplitude, frequency, time period, and energy are standard in every board paper.

Important Topics to Focus On

• Definition and equation of SHM: [latex]a = -\omega^2 x[/latex]
• Derivation of expressions for kinetic energy, potential energy, and total energy in SHM
• Period of vertical oscillation of a mass on a spring: [latex]T = 2\pi\sqrt{\frac{m}{k}}[/latex]
• Derivation of period of simple pendulum: [latex]T = 2\pi\sqrt{\frac{l}{g}}[/latex]
• Differences between free, damped, and forced oscillations
• Resonance — definition, condition, and real-life examples
• Numerical problems on SHM energy, period, frequency, and amplitude

Connection to Other Chapters

Periodic Motion connects directly to Chapter 6 (Wave Motion) and Chapter 7 (Mechanical Waves) in Grade 12, where oscillating particles collectively produce waves. The concept of resonance reappears in Chapter 8 (Waves in Pipes and Strings) and Chapter 19 (Alternating Currents), where electrical resonance in LC circuits mirrors mechanical resonance perfectly. Mastering this chapter makes those connections natural and intuitive.