The complete lecture — every idea comes alive in the live 3D panel on the right as you read. Scroll down; the animation keeps pace, and you can swing the pendulum and stretch the spring yourself.
1 — Oscillatory motion & basic terms
A body that moves to and fro about a fixed mean position, repeating its path at regular intervals, is oscillating. The swinging bob on the right is the classic example.
- Amplitude (x₀) — the maximum displacement from the mean position.
- Time period (T) — time for one complete vibration (a full round trip). Frequency (f) — vibrations per second; f = 1/T, unit hertz.
- Angular frequency — ω = 2πf = 2π/T (rad/s).
Exam point: one vibration = mean → extreme → mean → other extreme → mean. Half a swing is only half a vibration.
2 — SHM: definition & conditions
Simple harmonic motion is oscillation in which the acceleration is proportional to the displacement from the mean position and is always directed towards it.
The defining conditiona ∝ −x or a = −ω² x
- A stable mean position must exist.
- A restoring force proportional and opposite to displacement: F = −kx (Hooke's law).
- Inertia carries the body past the mean position; friction must be negligible.
The minus sign means the force always "disagrees" with the displacement — pull right, it tugs left — which is what drags the body back to the middle every time.
3 — Displacement, velocity & acceleration
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 = −ω² x
| Quantity | At mean (x = 0) | At extreme (x = ±x₀) |
|---|
| Velocity | max, v₀ = x₀ω | zero |
| Acceleration | zero | max, a₀ = x₀ω² |
A playground swing is fastest at the bottom of its arc and momentarily still at the top of each side — exactly this table in action.
4 — 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 (≈ −mg·x/L) is the restoring force, so the bob performs SHM.
Time periodT = 2π √(L / g)
- T depends only on length and g — not on the mass of the bob.
- T is independent of amplitude (small angles) — isochronism, the secret of pendulum clocks.
- To double T, make the string four times longer (T ∝ √L).
- The seconds pendulum (T = 2 s) is ≈ 0.99 m long on Earth.
5 — Practical: finding g with a pendulum
Rearranging the period formula gives the working formula of the standard BIEK practical:
Working formulag = 4π² L / T²
- Measure L to the centre of the bob (string + bob radius).
- Small swing (< 10°), released without a push.
- Time 20 complete vibrations and divide by 20 — this shrinks reaction-time error.
- Repeat for several lengths; plot L vs T² (straight line through origin); g = 4π² × slope.
Common errors: pushing at release, large amplitude, counting half-swings, measuring L only to the top of the bob.
6 — Hooke's law & the mass–spring system
Within the elastic limit a spring obeys Hooke's law, F = −kx, where k (N/m) is the spring constant. With F = ma this gives a = −(k/m)x — SHM with ω² = k/m.
Time period of mass–springT = 2π √(m / k) · f = (1/2π) √(k / m)
Real life: a car's suspension is a mass–spring system. Load it with passengers (bigger m) and the bounce slows down; fit stiffer springs (bigger k) and it vibrates faster. Amplitude changes T not at all.
7 — Energy interchange (K.E. ⇌ P.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 | zero |
| Extreme (±x₀) | zero | maximum |
On a swing, the highest points hold all the energy as height (P.E.); rushing through the bottom it is all motion (K.E.). Total energy ∝ amplitude squared.
8 — Free & forced oscillations, resonance
- Free oscillations — at the body's own natural frequency (a struck tuning fork, a plucked sitar string).
- Forced oscillations — driven by an external periodic force at the force's frequency.
- Resonance — driving frequency = natural frequency → amplitude grows dramatically.
- Damping — friction drains energy; shock absorbers damp a car's bounce on purpose.
Examples: timed pushes build a huge swing; wind drove the Tacoma Narrows Bridge (1940) at its natural frequency until it collapsed; soldiers break step on bridges; a radio tunes by resonating its circuit with one station.
9 — Worked numericals
pendulum period
A simple pendulum is 1 m long (g = 9.8 m/s²). Find T.
T = 2π√(L/g) = 2π√(1/9.8) = 2.0 s — the seconds pendulum.
finding g
A 90 cm pendulum completes 20 vibrations in 38.0 s. Find g.
T = 38.0/20 = 1.9 s
g = 4π²L/T² = 4 × 9.87 × 0.90 / 3.61 = 9.84 m/s²
mass–spring
A 0.5 kg block hangs on a spring with k = 200 N/m. Find T.
T = 2π√(m/k) = 2π√(0.5/200) = 2π × 0.05 = 0.314 s
10 — Exam recap
- SHM: a = −ω²x — acceleration ∝ displacement, towards the mean position.
- Hooke's law restoring force: F = −kx.
- 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: time 20 vibrations, g = 4π²L/T² (or 4π² × slope of L–T² graph).
- Mass–spring: T = 2π√(m/k).
- Energy: E = ½kx₀² constant; K.E. ⇌ P.E. interchange.
- Resonance: driving frequency = natural frequency → huge amplitude.