This MCQ module is based on: Periodic Motion Shm
TOPIC 24 OF 33
Periodic Motion Shm
🎓 Class 11
Physics
CBSE
Theory
Ch 13 – Oscillations
⏱ ~14 min
🌐 Language: [gtranslate]
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Periodic Motion Shm
13.1 Introduction — Repetition in Nature
Many motions around us repeat themselves — a pendulum swinging, a child on a swing, the tides rising and falling, the strings of a guitar vibrating, atoms in a crystal jiggling, the heart beating. Such periodic motion is everywhere in physics, from sub-atomic vibrations (10¹⁵ Hz) to galactic rotations (10⁻¹⁵ Hz).
A motion is oscillatory if the body moves to-and-fro about a fixed equilibrium position. Every oscillation is periodic, but not every periodic motion is oscillatory (a planet's orbit is periodic but not oscillatory).
13.2 Period and Frequency
Period T: the smallest time interval after which the motion repeats itself. SI unit: second.
Frequency ν (or f): number of repetitions per unit time. ν = 1/T. SI unit: hertz (Hz). 1 Hz = 1 cycle/s.
Angular frequency ω: ω = 2πν = 2π/T. SI unit: rad/s.
Frequency ν (or f): number of repetitions per unit time. ν = 1/T. SI unit: hertz (Hz). 1 Hz = 1 cycle/s.
Angular frequency ω: ω = 2πν = 2π/T. SI unit: rad/s.
| Phenomenon | Period T | Frequency ν |
|---|---|---|
| Heartbeat (resting) | ~ 0.85 s | ~ 1.2 Hz |
| Pendulum 1 m long | 2.0 s | 0.5 Hz |
| Tuning fork (A note) | 2.27 ms | 440 Hz |
| FM radio wave | 10⁻⁸ s | ~ 100 MHz |
| Caesium clock transition | 108.8 ps | 9.19 GHz |
| Earth's rotation | 86 400 s | 1.16 × 10⁻⁵ Hz |
13.3 Displacement, Amplitude and Phase
For oscillatory motion, the deviation of the particle from its equilibrium position is called the displacement x(t). The maximum magnitude of x is the amplitude A. The motion is bounded between −A and +A.
13.4 Simple Harmonic Motion (SHM)
The simplest, most universal kind of oscillation: a particle is pulled back to equilibrium by a force proportional to its displacement and directed oppositely:
Defining condition for SHM:
\[\boxed{\,F(x) = -k x \quad\Longleftrightarrow\quad a(x) = -\omega^2 x\,}\]
where ω² = k/m. Whenever the restoring force is linear in displacement, the motion is simple harmonic.
The equation of motion is therefore:
m d²x/dt² = −k x ⇒ d²x/dt² + ω² x = 0, ω = √(k/m)
The general solution is sinusoidal:
x(t) = A cos(ωt + φ)
Here A = amplitude, ω = angular frequency, φ = phase constant. Three pieces of information determine the motion: how big (A), how fast (ω), and where it starts (φ).
The argument (ωt + φ) is called the phase of the motion. φ alone is the phase at t = 0 (initial phase or epoch). Two SHMs with the same ω but different φ are said to have a phase difference — physically meaning one starts later than the other.
13.4.1 Why SHM Matters
Almost any system near a stable equilibrium behaves like an SHM oscillator for small displacements. Reason: any smooth potential V(x) with a minimum at x = 0 can be Taylor-expanded as V(x) ≈ V(0) + ½ V''(0) x², giving a linear restoring force F = −V'(x) = −V''(0) x. Hence atomic vibrations, electrical circuits, swinging buildings, water waves, even the field oscillations of light — all approximate SHM at small amplitudes. SHM is the local language of all oscillation.
Worked Examples
Example 13.1: Period and frequency from given ω
For SHM with x(t) = 0.05 cos(8πt + π/4) m, find amplitude, angular frequency, period, frequency, and initial phase.
A = 0.05 m; ω = 8π rad/s; T = 2π/ω = 2π/(8π) = 0.25 s; ν = 1/T = 4 Hz; initial phase φ = π/4 rad.
At t = 0, x(0) = 0.05 cos(π/4) = 0.05 × 0.707 = 0.0354 m.
At t = 0, x(0) = 0.05 cos(π/4) = 0.05 × 0.707 = 0.0354 m.
Example 13.2: Identifying SHM
Which of these motions are SHM? (a) x(t) = 5 sin(3t) (b) x(t) = 4 sin(2t) + 3 cos(2t) (c) x(t) = 7 sin²(πt) (d) x(t) = 6 + 2 cos(πt).
(a) SHM with ω = 3, A = 5, φ = π/2.
(b) SHM: 4 sin(2t) + 3 cos(2t) = 5 sin(2t + φ) with φ = arctan(3/4); ω = 2.
(c) Periodic, not SHM: sin²(πt) = (1 − cos(2πt))/2 — sinusoid plus a constant offset, not SHM about origin (but is SHM about x = 7/2 with ω = 2π).
(d) SHM about x₀ = 6, ω = π, A = 2.
(b) SHM: 4 sin(2t) + 3 cos(2t) = 5 sin(2t + φ) with φ = arctan(3/4); ω = 2.
(c) Periodic, not SHM: sin²(πt) = (1 − cos(2πt))/2 — sinusoid plus a constant offset, not SHM about origin (but is SHM about x = 7/2 with ω = 2π).
(d) SHM about x₀ = 6, ω = π, A = 2.
Example 13.3: Two oscillators, same ω, different phase
Two oscillators have x_1(t) = 3 cos(2t) and x_2(t) = 3 cos(2t − π/3). Find their phase difference and the time lag of x_2 behind x_1.
Phase difference Δφ = 0 − (−π/3) = π/3 rad. Time lag = Δφ/ω = (π/3)/2 = π/6 ≈ 0.524 s. Oscillator 2 reaches each maximum 0.524 s after oscillator 1.
Interactive: SHM Time-Graph Viewer L3 Apply
Drag A, ω and φ to see x(t) = A cos(ωt + φ) live on the graph.
Activity 13.1 — Periodic versus SHM Hunt
L4 Analyse
Predict: Of these motions — earth's orbit, ticking clock pendulum, beating heart, blinking traffic light, bouncing ball — which are SHM, which are merely periodic, and which are neither?
- For each motion, ask: does it return to the same state after a fixed T? (Periodic test.)
- If yes, ask: is the restoring force proportional to displacement from equilibrium? (SHM test.)
Earth's orbit: periodic (T ≈ 365 d), not SHM (force ∝ 1/r² inward, not a linear restoring force).
Pendulum (small swing): periodic and approximately SHM.
Heartbeat: approximately periodic, definitely not SHM (complex pulse shape).
Blinking light: periodic (square-wave), not SHM.
Bouncing ball: approximately periodic on a hard floor (decaying), not SHM (impulsive contact, not linear force).
Pendulum (small swing): periodic and approximately SHM.
Heartbeat: approximately periodic, definitely not SHM (complex pulse shape).
Blinking light: periodic (square-wave), not SHM.
Bouncing ball: approximately periodic on a hard floor (decaying), not SHM (impulsive contact, not linear force).
Competency-Based Questions
A particle moves so that x(t) = 0.04 sin(πt + π/3) m, where t is in seconds.
Q1. L1 Remember Define period and frequency of a periodic motion.
Period T is the smallest time interval after which the motion repeats; frequency ν = 1/T is the number of repetitions per unit time. ν is measured in hertz (Hz).
Q2. L2 Understand Why is a uniformly rotating object not SHM, even though it is periodic?
In SHM the body oscillates to-and-fro along a straight line about a fixed equilibrium under a linear restoring force. Uniform circular motion has constant speed along a circle and zero "restoring force along x" — it is periodic but not oscillatory in 1D.
Q3. L3 Apply Find A, ω, T, ν and initial phase for the given x(t).
A = 0.04 m; ω = π rad/s; T = 2π/ω = 2 s; ν = 0.5 Hz; initial phase = π/3 rad. At t = 0, x = 0.04 sin(π/3) = 0.04 × √3/2 = 0.0346 m.
Q4. L4 Analyse The graph of x(t) is shifted to the left by 0.5 s. What new phase must be used so that the same x(t) describes the shifted curve?
Shifting left by 0.5 s replaces t by (t + 0.5): x(t) = 0.04 sin(π(t + 0.5) + π/3) = 0.04 sin(πt + π/2 + π/3) = 0.04 sin(πt + 5π/6). New initial phase = 5π/6 rad.
Q5. L5 Evaluate A student says that x(t) = A cos(ωt) and x(t) = A sin(ωt) describe the same SHM. Is this true? Justify with an example.
Both describe SHM with the same A and ω but different initial phases (cos starts at maximum at t = 0; sin starts at zero, moving up). They differ by a phase of π/2. So they describe two SHMs of the same type but distinguishable by their initial conditions.
Assertion-Reason Questions
Assertion (A): Every SHM is periodic.
Reason (R): SHM is described by x(t) = A cos(ωt + φ), which has period 2π/ω.
Answer: A. Cosine is periodic with period 2π, so x(t) is periodic with period 2π/ω. R explains A.
Assertion (A): Every periodic motion is SHM.
Reason (R): All periodic motions repeat in equal intervals.
Answer: D. A is false — counter-example is uniform circular motion (periodic but not SHM, since it isn't to-and-fro along a line). R is true.
Assertion (A): Two SHMs of identical amplitude and frequency but with phase difference π are said to be "out of phase".
Reason (R): A phase shift of π corresponds to half a period in time.
Answer: A. Phase π = ωT/2 ⇒ time lag T/2. The two oscillators are exactly opposite at every instant.
Frequently Asked Questions - Periodic Motion Shm
What is the main concept covered in Periodic Motion Shm?
In NCERT Class 11 Physics Chapter 13 (Oscillations), "Periodic Motion Shm" covers core principles and equations needed for board exam success. The MyAiSchool lesson explains the topic with definitions, derivations, worked examples, and interactive simulations. Key formulas and dimensional analysis are included to build conceptual depth and problem-solving skills aligned with the CBSE 2025-26 syllabus.
How is Periodic Motion Shm useful in real-life applications?
Real-life applications of Periodic Motion Shm from NCERT Class 11 Physics Chapter 13 include engineering design, satellite mechanics, sports biomechanics, transportation safety, and electrical/electronic devices. The MyAiSchool lesson links every concept to a tangible example so students see physics as a problem-solving framework for the physical world, not as abstract formulas.
What are the key formulas in Periodic Motion Shm?
Key formulas in Periodic Motion Shm (NCERT Class 11 Physics Chapter 13 Oscillations) are derived step-by-step in the MyAiSchool lesson. Students should memorize the final formula AND understand its derivation for full board marks. Each formula is listed with its dimensional formula, SI unit, applicability range, and common pitfalls. The Summary section at the end of each part includes a quick-reference formula card.
How does this part connect to other parts of Chapter 13?
NCERT Class 11 Physics Chapter 13 (Oscillations) is structured so each part builds on the previous one. Periodic Motion Shm connects directly to neighbouring parts via shared definitions, units, and methodology. The MyAiSchool lesson cross-references related concepts with internal links so students can navigate the whole chapter as one connected story rather than disconnected fragments.
What types of CBSE board questions come from Periodic Motion Shm?
CBSE board questions from Periodic Motion Shm typically include: (1) 1-mark MCQs on definitions and formulas, (2) 2-mark short-answer derivations or applications, (3) 3-mark numerical problems with units, (4) 5-mark long-answer derivations followed by application. The MyAiSchool lesson tags each Competency-Based Question (CBQ) with Bloom level (L1-L6) so students know how to study for each weight.
How can students use the interactive simulation effectively?
The interactive simulation in the Periodic Motion Shm lesson allows students to adjust input parameters (sliders or selectors) and see physical quantities update in real time. To use it effectively: (1) try extreme values to understand limiting cases, (2) compare with the analytical formula, (3) check unit consistency, (4) test special configurations from worked examples. The simulation reinforces conceptual intuition that pure formula manipulation cannot.
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