Oscillations

A motion that repeats itself after equal intervals of time is periodic motion; the smallest such interval is the time period T. The number of repetitions per second is the frequency ν = 1/T, measured in hertz (Hz).

A periodic motion that moves back and forth about a fixed mean (equilibrium) position is called oscillatory or vibratory motion — a swinging pendulum, a block on a spring, the atoms in a molecule. Every oscillatory motion is periodic, but the reverse is not always true: uniform circular motion is periodic (it repeats every T) but it is not oscillatory because the particle does not move to and fro about one point.

At the mean position, the restoring force and hence the acceleration are zero and speed is maximum; at the extreme positions (turning points) velocity is zero and the restoring force/acceleration is maximum, directed back towards the mean position. This restoring force is what makes an oscillation possible — remove it and the motion stops repeating.

Simple harmonic motion is the special periodic motion in which the restoring force (and acceleration) is directly proportional to the displacement from the mean position and is always directed towards it: F = −kx. This single condition is enough to derive the sinusoidal shape of the motion.

The general solution is x(t) = A cos(ωt + φ), where A is the amplitude (maximum displacement from mean position), ω is the angular frequency, and (ωt + φ) is the phase at time t; φ is the phase constant, fixed by the position and velocity at t = 0. Since cosine repeats every 2π, the motion repeats when ωt increases by 2π, giving T = 2π/ω.

SHM can equally be written using sine — x(t) = A sin(ωt + φ) — the two differ only by a phase of π/2 and by the choice of where t = 0 is taken. Both x = A cos ωt (starting at extreme position) and x = A sin ωt (starting at mean position) are common starting conditions in numericals.

SHM can be visualised as the projection, on any diameter, of a particle moving with uniform speed on a reference circle of radius A. If the particle P moves on the circle with constant angular speed ω, its foot of perpendicular on the x-axis (or y-axis) executes SHM of amplitude A and angular frequency ω — this is exactly why ω in SHM is called angular frequency even though nothing is physically rotating.

This picture is useful for reading off phase quickly: at t = 0 the reference particle is at angle φ from the axis of projection, and as it goes around, the projection point sweeps between +A and −A, matching x(t) = A cos(ωt + φ).

Do not confuse this with real circular motion — the connection is only geometrical, used as a tool to visualise phase and to derive v(t) and a(t) using simple circular-motion relations (v = Aω, centripetal acceleration = Aω²).

Differentiating x(t) = A cos(ωt + φ) gives velocity and acceleration.

v(t) = dx/dt = −Aω sin(ωt + φ). Velocity is maximum (= Aω) at the mean position (x = 0) and zero at the extreme positions (x = ±A). Velocity is 90° (π/2) out of phase with displacement.

a(t) = dv/dt = −Aω²cos(ωt + φ) = −ω²x. Acceleration is directly proportional to displacement and always opposite in sign — it always points towards the mean position. Acceleration is maximum (= Aω²) at the extremes and zero at the mean position, and it is 180° out of phase with displacement (exactly opposite) but 90° out of phase with velocity.

The relation v = ω√(A² − x²) (obtained by eliminating t between x and v) is very useful for numericals that give displacement and ask for speed, or vice versa, without needing time.

From Newton's second law, F = ma = −mω²x = −kx, where k = mω² is called the force constant (or spring constant when the system is literally a spring). This is Hooke's-law-like behaviour: the restoring force is proportional to displacement and opposite to it.

Conversely, whenever the net force (or torque, for angular oscillations) on a system can be written as F = −kx for small displacements from equilibrium, the resulting motion is guaranteed to be SHM with ω = √(k/m). This is the standard method used in the chapter to prove that a spring–mass system, a simple pendulum (for small angles) and a floating cylinder all execute SHM — write down the restoring force/torque, show it is linear in displacement, and read off ω.

Note k has units N/m and depends on the physical system (stiffness of spring, weight and length for a pendulum, etc.), while ω = √(k/m) combines k with the inertia (mass) of the oscillating body.

SHM conserves total mechanical energy because the restoring force is conservative. At displacement x, kinetic energy K = ½mv² = ½mω²(A² − x²), and potential energy U = ½kx² = ½mω²x² (taking U = 0 at the mean position). Both K and U oscillate with time but their sum stays constant.

Total energy E = K + U = ½mω²A² = ½kA², independent of x and of time — it depends only on amplitude, mass and ω (or k). This is one of the most tested numerical ideas: E is fixed by A and k alone; at any displacement x, K = E − U so you can find one from the other without extra data.

K is maximum (= E) at the mean position where v is maximum; U is maximum (= E) at the extreme positions where x = ±A. Both K and U vary as cos²/sin² of (ωt + φ), so each oscillates at twice the frequency of the displacement itself, always staying non-negative.

Spring–mass system: A block of mass m on a frictionless surface attached to a spring of force constant k, displaced by x, feels restoring force F = −kx, so ω = √(k/m) and T = 2π√(m/k). This holds whether the spring is horizontal or vertical (gravity only shifts the mean position, it does not change T).

For springs in series, 1/k_eff = 1/k₁ + 1/k₂ (effective spring is softer, larger T); in parallel, k_eff = k₁ + k₂ (stiffer, smaller T).

Simple pendulum: a point mass m on a light, inextensible string of length L. For angular displacement θ, restoring torque about the support is τ = −mgL sin θ; for small θ, sin θ ≈ θ, giving τ = −mgLθ, i.e. angular SHM with ω² = g/L (moment of inertia I = mL², and mgLθ = Iα gives α = −(g/L)θ). Hence T = 2π√(L/g), independent of amplitude (for small θ) and of the mass of the bob — this is exactly why it is used to measure g experimentally.

Real oscillators lose energy to friction or a resistive medium, so amplitude decreases with time — this is damped oscillation. For a block on a spring in a viscous medium, besides the spring force −kx there is a damping force proportional to velocity, −bv (b = damping constant), so the equation of motion is m(d²x/dt²) = −kx − b(dx/dt).

The solution (for small damping) is x(t) = A e^−bt/2m cos(ω′t + φ), where the amplitude A e^−bt/2m decays exponentially with time, and the (reduced) angular frequency is ω′ = √(k/m − b²/4m²), slightly less than the undamped ω₀ = √(k/m).

Mechanical energy is not conserved in damped SHM: E(t) = ½kA²e^−bt/m, decaying exponentially. Larger b means faster decay; if damping is too large, the system does not oscillate at all (overdamped) and just returns slowly to equilibrium — car shock absorbers are deliberately designed this way (critically/heavily damped).

When an external periodic force F(t) = F₀ cos(ω_dt) is applied to a damped oscillator (natural frequency ω₀), after the initial transients die out the system settles into steady oscillations at the driving frequency ω_d, not its own natural frequency — this is forced (driven) oscillation.

The steady-state amplitude depends on how close ω_d is to ω₀ and on the damping: it is largest when ω_d is close to ω₀. When ω_d = ω₀ (or very close to it, for small damping), the amplitude becomes maximum — this condition is called resonance. With very small damping the resonance amplitude can become very large (in the ideal undamped case it would grow without bound); real damping always keeps it finite.

Everyday examples: pushing a swing at its own natural rhythm builds up a large amplitude; soldiers break step crossing a bridge to avoid matching the bridge's natural frequency; a radio tuning circuit is adjusted (LC resonance) to match the frequency of the desired broadcast station.