The full, readable lecture — what makes a motion "simple harmonic", the equations for displacement, velocity and acceleration, the simple pendulum and how it measures g, the mass–spring system with Hooke's law, energy interchange, and resonance. As you scroll, the panel on the right plays out each idea with an everyday object you already know — a swing, a car's spring, a clock, a sitar string.
1 — Oscillatory motion & basic terms
A body that moves to and fro about a fixed mean position, repeating its path at regular intervals, is said to oscillate (or vibrate). A child on a playground swing is the perfect example — it rides out to one side, falls back through the bottom, rises to the other side, and returns, over and over.
- Mean (equilibrium) position — the point where the net force is zero; the swing hangs there at rest if undisturbed.
- Displacement (x) — the distance from the mean position at any instant, with direction. A vector.
- Amplitude (x₀) — the maximum displacement from the mean position (the highest the swing reaches).
- Time period (T) — the time for one complete vibration (one full round trip). Unit: second (s).
- Frequency (f) — vibrations completed per second. Unit: hertz (Hz). f = 1/T.
- Angular frequency (ω) — ω = 2πf = 2π/T. Unit: rad/s.
Exam point: one vibration means a complete round trip — mean → extreme → mean → other extreme → mean. Half a swing is only half a vibration.
2 — SHM: definition & the restoring force
Simple harmonic motion (SHM) is oscillatory motion in which the acceleration is directly proportional to the displacement from the mean position and is always directed towards it. A car driving over a speed-breaker shows it cleanly: the bump compresses the suspension spring, the spring pushes the body back up, it overshoots, and the car bobs up and down about its rest height.
The defining conditiona ∝ −x or a = −ω² x
(the minus sign means a always points back towards the mean position)
- A stable equilibrium (mean) position must exist (the car's rest height).
- A restoring force proportional and opposite to displacement: F = −kx (Hooke's law).
- Inertia carries the body past the mean position so it keeps oscillating; friction must be small.
Why the minus sign? Push the body right and the force pulls left; push it down and the spring pushes up. The force always "disagrees" with the displacement — that is what drags the body back to the middle every time.
3 — The simple pendulum
A simple pendulum is a small heavy bob on a light, inextensible string. Displaced through a small angle (≤ 10°), the component of gravity along the arc, F = −mg sin θ ≈ −mg(x/L), is the restoring force, so the bob performs SHM. This is the heart of every pendulum clock — the steady tick that kept the world on time for three centuries.
Time period of a simple pendulumT = 2π √(L / g)
L = length to the centre of the bob · g = 9.8 m/s²
- T depends only on length L and g — not on the mass of the bob.
- T is independent of amplitude for small angles — the law of isochronism behind clocks.
- T ∝ √L: to double the period, make the string four times longer.
- The seconds pendulum (T = 2 s) is L = gT²/4π² ≈ 0.99 m long on Earth.
Practical (BIEK): rearranging gives g = 4π²L / T². Time 20 complete vibrations (to shrink reaction-time error), find T, then compute g — or plot L against T² and use g = 4π² × slope. Common errors: pushing at release, large amplitude, counting half-swings.
4 — Displacement, velocity & acceleration
Pluck a sitar or guitar string and let go: it whips back and forth about its straight rest line, and a single point on it traces the cleanest SHM of all. Starting from the mean position at t = 0 the motion is sinusoidal:
Equations of SHMx = x₀ sin ωt
v = x₀ ω cos ωt = ω √(x₀² − x²)
a = −x₀ ω² sin ωt = −ω² x
| Quantity | At mean (x = 0) | At extreme (x = ±x₀) |
|---|
| Velocity | maximum, v₀ = x₀ω | zero |
| Acceleration | zero | maximum, a₀ = x₀ω² |
| Restoring force | zero | maximum |
Notice the see-saw pattern: where the speed is greatest the acceleration vanishes, and where the string momentarily halts (the extremes) the pull back to the centre is strongest. A swing is fastest at the bottom and still for an instant at the top of each side — exactly this table.
5 — Hooke's law & the mass–spring system
The cleanest model of SHM is a mass m on a spring of spring constant k. Within the elastic limit the spring obeys Hooke's law, F = −kx. A struck tuning fork is exactly this: each prong is a tiny stiff spring with mass, vibrating at one pure frequency set by its own k and m.
Time period of a mass–spring systemT = 2π √(m / k) · f = (1/2π) √(k / m)
| Change made | Effect on T = 2π√(m/k) |
|---|
| Heavier mass (m × 4) | T doubles — a loaded car bounces more slowly |
| Stiffer spring (k × 4) | T halves — sports suspensions vibrate faster |
| Bigger amplitude | No effect — T is independent of amplitude |
Real life: a car's suspension is exactly this system — more passengers (bigger m) → a slower, softer bounce; stiffer springs (bigger k) → a faster vibration. Shorten a tuning fork's prongs and it rings at a higher pitch for the same reason.
6 — Energy interchange (K.E. ⇌ P.E.)
In SHM the total mechanical energy is constant (no friction), but it continuously converts between kinetic and potential form, twice per vibration each way. On a swing the highest points hold all the energy as height (P.E.); rushing through the bottom it is all motion (K.E.).
Energy at displacement xP.E. = ½ k x² · K.E. = ½ k (x₀² − x²)
Total E = ½ k x₀² = constant
| Position | K.E. | P.E. |
|---|
| Mean (x = 0) | maximum (= total E) | zero |
| Extreme (±x₀) | zero | maximum (= total E) |
| At x = x₀/√2 | half of E | half of E |
Exam point: total energy in SHM is proportional to the square of the amplitude (E ∝ x₀²) and to the square of the frequency. Energy just sloshes back and forth like water in a tilting bottle — the total never changes.
7 — Free & forced oscillations, resonance
- Free oscillations — a body vibrating on its own at its natural frequency after one disturbance (a struck tuning fork, a plucked sitar string).
- Forced oscillations — a body driven by an external periodic force, at that force's frequency.
- Resonance — when the driving frequency equals the natural frequency, the amplitude grows dramatically.
- Damping — friction steadily drains the energy so amplitude dies away (shock absorbers damp a car's bounce on purpose).
Pushing a swing: small pushes timed to the swing's own rhythm build an enormous amplitude — push off-beat and it barely moves. The Tacoma Narrows Bridge (1940) oscillated when wind supplied periodic forces near its natural frequency until it tore itself apart; soldiers break step on bridges for the same reason. Useful resonance tunes a radio and powers a microwave oven.
8 — Worked numericals & exam recap
simple pendulum — time period
Find T and f of a simple pendulum 1 m long (g = 9.8 m/s²).
T = 2π√(L/g) = 2π√(1/9.8) = 2.0 s — the seconds pendulum.
f = 1/T = 0.5 Hz
finding g from a pendulum
A 90 cm pendulum completes 20 vibrations in 38.0 s. Find g.
T = 38.0/20 = 1.9 s · L = 0.90 m
g = 4π²L/T² = 4 × 9.87 × 0.90 / 3.61 = 9.84 m/s²
mass–spring system
A 0.5 kg block hangs on a spring with k = 200 N/m. Find T and f.
T = 2π√(m/k) = 2π√(0.5/200) = 2π × 0.05 = 0.314 s
f = 1/T = 3.18 Hz
- SHM: a = −ω²x — acceleration ∝ displacement, towards the mean position.
- Hooke's law restoring force: F = −kx (k in N/m).
- x = x₀ sin ωt; v max at mean, a max at extremes.
- Pendulum: T = 2π√(L/g) — mass and amplitude don't matter; find g via g = 4π²L/T².
- Mass–spring: T = 2π√(m/k).
- Energy: E = ½kx₀² constant; K.E. ⇌ P.E.; E ∝ x₀².
- Resonance: driving frequency = natural frequency → huge amplitude.