NEB Class 12 • Physics • Mechanics

Periodic Motion: NEB Class 12 Physics Guide

Treat an oscillator as one system described simultaneously by force, differential equation, displacement, phase, velocity, acceleration, energy and experiment.

Periodic motion versus SHM
Spring, pendulum, phase and energy
Damping, resonance, graphs and practical method

Force, motion, energy and graphs must describe the same oscillator.

Curriculum boundary

Periodic motion in the Grade 12 sequence

The CDC Grade 12 Physics page and secondary curriculum place periodic motion after rotational dynamics and include oscillatory terminology, SHM, projection/circular-motion connection, energy, spring and pendulum models, damping and forced oscillation/resonance at the stated school depth. Follow the current school scheme for the exact assessment sequence.

Prerequisites include graphs, radians, circular motion, Hooke’s law, energy and the rotation language of angular frequency. Ordinary frequency f in hertz and angular frequency ω in rad s⁻¹ are related by ω=2πf=2π/T.

Core language

Distinguish periodic, oscillatory and harmonic motion

Periodic motion repeats its state after equal time intervals. Oscillatory motion moves back and forth about an equilibrium configuration. Not every periodic motion is an oscillation: uniform circular motion is periodic but does not move back and forth along the circle. Not every oscillation is exactly simple harmonic: large-angle pendulum motion is periodic/oscillatory but departs from the small-angle SHM model.

One complete cycle returns position and direction of motion to the starting state. Period T is time per cycle; frequency f is cycles per second; amplitude A is maximum displacement from equilibrium. Displacement x has sign, while distance travelled does not. Phase describes position within the cycle and is measured as an angle.

Cycle check

Starting at x=A and moving toward equilibrium, the oscillator reaches equilibrium after T/4, x=−A after T/2, equilibrium after 3T/4 and returns to x=A with the original direction/state after T.

Defining condition

Simple harmonic motion requires a linear restoring acceleration

SHM satisfies a∝−x, written a=−ω²x. The minus sign says acceleration points toward equilibrium. From Newton’s second law, the net restoring force is F=−mω²x. The constant ω² sets how quickly the system responds. The OpenStax SHM treatment connects this condition to the equation of motion.

A common solution is x=Acos(ωt+φ). Differentiating gives v=−Aωsin(ωt+φ) and a=−Aω²cos(ωt+φ)=−ω²x. Maximum speed Aω occurs at equilibrium; maximum acceleration Aω² occurs at the extremes. At an extreme, speed is zero but acceleration is not zero.

Worked state example

For x=0.080cos(4t) m, A=0.080 m, ω=4 rad s⁻¹, T=2π/4=1.57 s and vmax=Aω=0.32 m s⁻¹. At t=0, x=A, v=0 and a=−Aω²=−1.28 m s⁻².

Representations

Use phase to align displacement, velocity and acceleration

The phase angle ωt+φ tells where the oscillator is in its cycle. Two oscillators with equal frequency can differ by phase Δφ. A quarter-cycle shift corresponds to π/2 radians; a half-cycle shift corresponds to π radians. Phase difference is meaningful only when the reference and sign convention are clear.

On an x–t graph, slope is velocity; on a v–t graph, slope is acceleration. At x maxima, x–t slope is zero. At equilibrium, |v| is maximum. The acceleration graph is the negative displacement graph scaled by ω². The area under v–t gives displacement change, while area under a–t gives velocity change.

SHM can be viewed as the projection of uniform circular motion: the projection of radius A onto an axis varies sinusoidally while the phase point moves with angular speed ω. This is a mathematical representation, not a claim that every spring mass travels in a hidden circle.

Graph audit

If x crosses zero with negative slope, velocity is maximum negative and acceleration is zero. Any proposed graph showing positive v there contradicts the slope. Use graph relationships before equations.

Spring–mass oscillator

Hooke’s law determines angular frequency

For an ideal horizontal mass m attached to spring constant k, F=−kx and ma=−kx. Comparing a=−(k/m)x with SHM gives ω=√(k/m), so T=2π√(m/k). Greater mass increases period; a stiffer spring decreases it. In the ideal linear model, period is independent of amplitude.

A vertical spring has a shifted equilibrium because mg stretches the spring. Measuring displacement from the new equilibrium leaves the same small-oscillation equation and period. Do not insert g into the final period merely because gravity establishes the equilibrium position.

Worked spring example

A 0.50 kg mass oscillates on k=200 N m⁻¹. ω=√(200/0.50)=20 rad s⁻¹ and T=2π/20=0.314 s. If mass becomes 2.0 kg with the same spring, period doubles because T∝√m and mass increases by factor four.

Explore parameter changes with the institutional PhET Masses and Springs simulation, then write predictions before moving sliders.

Simple pendulum

The small-angle approximation creates SHM

A pendulum bob has tangential restoring force −mg sinθ. For small angles in radians, sinθ≈θ and arc displacement s=Lθ, producing approximately a=−(g/L)s. Thus ω=√(g/L) and T=2π√(L/g). Mass cancels. The ideal result assumes a point-like bob, light inextensible string, small angle, fixed support and negligible resistance.

At larger amplitude, sinθ is not equal to θ and period depends weakly on amplitude, so the motion is no longer exact SHM. Do not quote a universal angle cutoff without context; state the approximation and follow the accuracy expected by the problem or experiment.

Worked pendulum example

For L=1.00 m and g=9.81 m s⁻², T=2π√(1/9.81)=2.01 s. If length becomes 0.25 m at the same place, period halves because T∝√L.

Use the OpenStax pendulum resource and PhET Pendulum Lab to test length, gravity and amplitude while recording controlled variables.

Energy picture

Kinetic and potential energy exchange through the cycle

For an ideal spring oscillator, total mechanical energy E=½kA². Elastic potential U=½kx² and kinetic K=E−U=½k(A²−x²). At extremes, K=0 and U=E. At equilibrium, U is minimum in the chosen reference and K is maximum. Energy remains constant only in the ideal undamped model.

Using v²=ω²(A²−x²), the energy and kinematic descriptions agree. A common mistake is to say acceleration is greatest where kinetic energy is greatest; actually acceleration magnitude is greatest at extremes while speed and kinetic energy are greatest at equilibrium.

Energy example

A spring k=80 N m⁻¹ has amplitude 0.10 m. Total energy is ½×80×0.10²=0.40 J. At x=0.060 m, U=0.144 J and K=0.256 J. The two add to 0.40 J.

Real oscillators

Damping removes energy; forcing can produce resonance

Damping from friction or resistance reduces amplitude and mechanical energy. Light damping allows many oscillations; critical damping returns toward equilibrium rapidly without oscillation; overdamping returns more slowly without overshoot. Precise classifications depend on the system model.

A forced oscillator receives periodic external input. Its steady response depends on driving frequency, natural frequency and damping. Resonance is a large response near a characteristic frequency; damping limits and broadens the response. Resonance can be useful in tuning and measurement but dangerous in structures or machines if not controlled.

Do not say resonance means “frequency becomes infinite.” The driving frequency is finite; idealised amplitude can become unbounded only in an unrealistic lossless model driven exactly at natural frequency indefinitely. Real systems have damping, nonlinearities and strength limits.

Push timing

A swing gains energy efficiently when pushes are timed with its motion. Random pushes can cancel or add little energy. The mechanism is phase-sensitive work, not merely “more force.”

Practical method

Measure a period without hiding uncertainty

For a pendulum, measure length to the bob’s centre, use small displacement, release without push, time many complete oscillations from the same reference and repeat. Divide total time by cycle count. Vary length while controlling amplitude and bob/setup, then test T²∝L with a graph.

Record raw timing, units, instrument resolution and repeats. Reaction time is a specific limitation. Timing many cycles reduces its fractional effect; a fiducial marker improves consistent crossing. Do not delete a reading because it seems inconvenient—investigate and document the decision.

For a spring, test T² against mass in the linear region and account for any effective spring mass only if taught and justified. Follow teacher-approved apparatus limits; do not overload springs or suspend unsafe masses.

Common exam mistakes: confusing f and ω, using degrees inside small-angle calculus relations, assuming every periodic motion is SHM, claiming pendulum period depends on bob mass, and forgetting that phase/state includes direction. Use the Class 12 revision roadmap for spaced retests.

For online or physical tuition, call 9846662070. The MKS Education panel below supports SAT/IELTS/PTE/DET and study-abroad pre-counselling after Grade 12.

Frequently asked questions

Questions students ask about Periodic Motion

Is every periodic motion simple harmonic?

No. SHM additionally requires acceleration proportional to displacement and directed toward equilibrium.

Where is speed maximum in SHM?

At equilibrium, where displacement is zero; speed is zero at the extreme positions.

Does pendulum mass affect its ideal small-angle period?

No. In the ideal simple-pendulum model, period depends on length and local gravitational acceleration.

Why is the vertical spring period independent of g?

Gravity shifts the equilibrium position, but displacement about that equilibrium obeys the same ideal spring equation.

What is resonance?

It is a large forced response near a system’s characteristic frequency, limited in real systems by damping and nonlinear effects.

Where can I get Periodic Motion tuition?

Call 9846662070 for current KTM Tuition online or physical NEB schedules and fees.

Checked sources

References and related learning

Review prerequisites in Circular Motion and connect this chapter to the complete Class 12 Physics plan. Curriculum and institutional pages were checked on 2 August 2026; follow current CDC, NEB and school notices if requirements change.

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