JEE/NEET Physics · Oscillations & Waves series · Part 1 of 8 · All parts →
- SHM = to-and-fro motion where the pull back is proportional to the distance pushed
- a = −ω²x: the further you displace, the stronger the restoring pull (opposite direction)
- One formula gives position at any time: x = A·sin(ωt)
- Time period T is INDEPENDENT of amplitude (for ideal SHM)
- ω (omega) = 2π/T — the motion’s ‘clock rate’ in radians per second
A pendulum clock, a guitar string, a car’s suspension, your heartbeat’s pacemaker cells — all run on one kind of motion: push away from rest, and a restoring pull proportional to your displacement brings you back, overshoot, repeat. That’s simple harmonic motion (SHM) — the engine of this entire series. Part 1 of the Oscillations & Waves series.
- The simple idea: the spring rule
- What each letter means
- The motion formula: x = A·sin(ωt)
- Why the period doesn’t care about amplitude
- Velocity and acceleration at the key points
- Solved examples
- Common mistakes
- This physics in your daily life
- Practice set
- Recap
The Simple Idea: The Spring Rule
Stretch a spring and it pulls back — the further you stretch, the harder it pulls (Hooke’s law, WEP Part 7). Now imagine a ball on such a spring, nudged once and left alone: it oscillates — pulled to centre, overshooting, pulled back, forever (no friction). Any motion with this ‘restoring pull ∝ displacement, opposite direction’ signature is SHM — pendulums (small swings), bobbing boats, vibrating atoms.
What Each Letter Means
| Letter | What it means (plain words) | Value / unit |
|---|---|---|
| x | displacement — how far from the centre (mid-point), right now | metres |
| A | amplitude — the MAXIMUM displacement (the size of the swing) | metres |
| T | time period — time for one full to-and-fro | seconds |
| f | frequency — full oscillations per second (f = 1/T) | hertz (Hz) |
| ω | angular frequency — the motion’s clock in radians/second (ω = 2π/T = 2πf) | rad/s |
| φ (phi) | phase — where in the cycle the motion started | radians (often 0 in problems) |
The Motion Formula: x = A·sin(ωt)
Where is the ball at time t? Multiply: amplitude × sine of (clock rate × time):
Read it: the sine waves between −1 and +1, so x waves between −A and +A. The clock ω converts seconds into radians; when ωt completes 2π, one full cycle ends — that’s why T = 2π/ω.
Why the Period Doesn’t Care About Amplitude
The magic property: pull it twice as far, the restoring pull is twice as strong — you travel double the distance at double the average speed. Both effects cancel: the period stays the same. Big swings, small swings (ideally): same clock. That’s why pendulum clocks keep time as the swing decays — and why musicians’ strings hold their pitch as notes fade.
Velocity and Acceleration at the Key Points
| Position | Speed | Acceleration | Why |
|---|---|---|---|
| Centre (x = 0) | MAXIMUM (v = Aω) | zero | all energy is motion |
| Extreme ends (x = ±A) | zero | MAXIMUM (a = ω²A) | all energy is stored — pull-back strongest |
Solved Examples
f = 1/T = 2 Hz. ω = 2πf = 4π ≈ 12.57 rad/s. v_max = Aω = 0.02 × 12.57 ≈ 0.25 m/s.
Check: small quick swing → modest top speed, sensible. ✔
Answer: f = 2 Hz; ω = 4π rad/s; v_max ≈ 0.25 m/s
Match with x = A sin(ωt): A = 5 cm, ω = 10 → T = 2π/10 ≈ 0.63 s.
At T/4: sin(ω·T/4) = sin(π/2) = 1 → x = +5 cm — the far positive end (speed zero). ✔
Answer: A = 5 cm; T ≈ 0.63 s; x(T/4) = +A
The SHM clock for a spring: T = 2π√(m/k) = 2π√(2/200) = 2π/10 ≈ 0.63 s.
Feel it: heavier = lazier clock (slower); stiffer spring = faster clock. Mass and stiffness decide everything; amplitude doesn’t appear at all — the magic property above. ✔
Answer: T ≈ 0.63 s, whatever the amplitude
- Forgetting the minus sign in a = −ω²x. The acceleration always points back to centre — that minus is the physics, not decoration.
- Using degrees in sin(ωt). ω is in rad/s and t in seconds — ωt is in radians. A 30° in place of π/6 slips every answer.
- Assuming amplitude affects the period. In ideal SHM it never does — bigger swing, same clock.
- Confusing frequency (Hz) with ω (rad/s). They differ by 2π. The formula sheet uses ω; the data sheet often gives f — convert first.
This Physics in Your Daily Life
- Every pendulum wall clock is SHM’s amplitude-independence at work — the swing decays all day, the tick rate never changes.
- Car suspension is a spring-mass SHM tuned so your car’s natural bounce sits away from road-shake frequencies — comfort is tuned SHM (Part 4 explains the tuning).
- Guitar/piano strings run SHM transversely: pitch = frequency, loudness = amplitude. Strum harder: louder, SAME pitch — amplitude-independence, heard.
- Atoms in molecules vibrate in near-SHM — infrared spectroscopy (finding chemicals, testing food purity) literally reads these vibration frequencies.
- Quartz watches: a tiny tuning-fork crystal oscillates at 32,768 Hz with SHM precision — your wristwatch is a physics lab.
Practice set (answers hidden — try first)
(NEET-level) T = 2 s. f and ω:
(JEE Main-level) x = 4 sin(2πt) cm. v_max:
(Concept) Doubling the amplitude of ideal SHM changes the period by:
(NEET-level) In SHM, acceleration is maximum where displacement is:
(JEE Main-level) A 1 kg mass on k = 100 N/m spring. Period:
- 🧠 Chant: ‘fastest at the centre, strongest pull at the ends’.
- 🧠 Amplitude-free clock: ‘big swing or small — same tick’.
- 🧠 2π converter: ‘ω to f, divide by 2π’.
- 🏠 Daily: a decaying guitar note keeps its pitch — SHM’s magic property, audible.
- 🏠 Daily: your quartz watch ticks on SHM at 32,768 Hz.
- 🔁 SHM: a = −ω²x, pull-back proportional to displacement
- 🔁 x = A sin(ωt); v = Aω at centre; a = ω²A at extremes
- 🔁 ω = 2π/T = 2πf
- SHM signature: restoring pull ∝ displacement, opposite direction
- x = A sin(ωt); v_max = Aω at centre; a_max = ω²A at ends
- T = 1/f; ω = 2πf = 2π/T — convert between them first
- period independent of amplitude (ideal SHM)
- spring-mass: T = 2π√(m/k)
Quick revision
- SHM = to-and-fro motion where the pull back is proportional to the distance pushed
- a = −ω²x: the further you displace, the stronger the restoring pull (opposite direction)
- One formula gives position at any time: x = A·sin(ωt)
- Time period T is INDEPENDENT of amplitude (for ideal SHM)
- ω (omega) = 2π/T — the motion’s ‘clock rate’ in radians per second
- The simple idea: the spring rule
Have a doubt on this topic?
