9478 / 2027
Oscillations overview

Topic 1 of 6

Describe and measure an oscillation

An oscillation repeats about an equilibrium position. To describe it, identify that position, measure displacement from it and count a complete cycle rather than every crossing of the centre.

Equilibrium and free oscillation

At the equilibrium position, the body can remain at rest with forces balanced when no periodic drive is applied. Displacement x is the signed distance from this position. A body can pass through equilibrium at nonzero speed; being at that position does not mean it is at rest.

A horizontal cart attached to a spring can oscillate after a displacement and release. In the ideal free model, no energy is gained from or lost to the environment: kinetic and potential energy exchange while their total remains constant. Restoring forces still act; there is no continuing periodic drive.

A small-angle pendulum is another approximate example when losses are negligible. Real carts and pendulums gradually lose mechanical energy unless supplied with more. Their damped responses are considered later in the chapter.

Measure displacement from equilibrium

The spring-cart is shown at one position, with positive x to the right. The front-mounted pointer reads 23.0 cm; the rest position reads 20.0 cm. This snapshot alone does not tell which way the cart is moving.

Ruler position, signed displacement and amplitude are differentA horizontal spring is fixed to a left-hand wall and attached to a cart on a low-friction track. The cart's front-mounted pointer lies above the 23-centimetre ruler mark. Its rest reference at 20 centimetres is shown by a dashed vertical line. Position ticks have one common scale, twenty drawing units per centimetre. The two possible extreme pointer readings, 15 and 25 centimetres, are marked on that same scale. A blue dimension arrow from 20 to 23 represents positive three centimetres of displacement; it is not a velocity or force arrow. A separate bracket from 20 to 25 gives amplitude five centimetres. The full extreme-to-extreme span is ten centimetres. The pointer is drawn in front of the track and ruler. This position snapshot is separate from the later model's chosen zero time.Equilibrium referencewhen at restCartPointer15202325cmExtremeExtremex = +3.0 cmx0 = 5.0 cm

The amplitude is 5.0 cm on either side of equilibrium. The 10.0 cm extreme-to-extreme distance is twice the amplitude.

This is a position snapshot: the ruler reads position, while displacement is measured from equilibrium. The two extremes are equally spaced about equilibrium. The snapshot does not specify a velocity direction or the time origin.

Amplitude, period and the two frequency quantities

The amplitude x0 is the maximum magnitude of displacement, in metres. It is non-negative. If a rest pointer is at 20.0 cm and the extremes are 15.0 cm and 25.0 cm, the amplitude is 5.0 cm = 0.050 m, not the 10.0 cm distance between extremes.

The period T is the time for one complete cycle, in seconds. The system must return to the same state of motion: two centre crossings in the same direction are one period apart. Consecutive crossings in opposite directions are half a period apart.

The frequency f is cycles per second, in hertz (Hz = s-1). The angular frequency ω describes the rate at which the sinusoidal phase advances, in rad/s. One cycle corresponds to 2π radians:

T = 1/f = 2π/ω
ω = 2πf

Frequency and angular frequency have different numerical values and units. If ω is in rad/s and t is in seconds, the phase ωt is in radians.

Estimate the scale before calculating

A few-centimetre amplitude and roughly 20 cycles in 30 s suggest a period of order 1 s, frequency of order 1 Hz and angular frequency of a few rad/s. These are rough scales, not a reason to make f and ω numerically equal.

Worked repeated-cycle timing

Average the time for complete cycles

Supplied stopwatch readings for 20 complete cycles in each trial
TrialTime for 20 cycles / s
131.2
231.4
331.6
Mean time = (31.2 + 31.4 + 31.6)/3 = 31.4 s
T = 31.4/20 = 1.57 s
f = 1/T = 0.637 Hz
ω = 2π/T = 4.00 rad/s

These illustrative supplied readings describe the scale of one oscillator. Timing only a rightward centre crossing to the next leftward crossing would measure T/2 instead.

Optional check An oscillator crosses equilibrium moving right, then crosses it moving left 0.785 s later. What is its period?
An oscillator crosses equilibrium moving right, then crosses it moving left 0.785 s later. What is its period?

Measure displacement and timing together

  1. Establish equilibrium. Let the undriven system settle and mark the rest reading. Calibrate a displacement scale in the plane of motion. Subtract the equilibrium reading from position to obtain signed x.
  2. Use a reproducible release. Keep the spring attachments and supports secure, remain within the apparatus range and release without a continuing push.
  3. Time a stated number of full cycles. Use crossings of the same reference in the same direction, then repeat. Choose a stopwatch range and resolution suitable for the full timing interval.
  4. Record position against time. A suitable position sensor or calibrated video can give an x-t record. Track one point, retain the sample or frame times, view perpendicular to the motion plane and calibrate distance in that plane.
  5. Check the model conditions. Compare successive cycles for changing period or amplitude. Keep mass and restoring arrangement fixed when comparing amplitudes, and remain within the linear or small-angle range.

Many-cycle timing reduces the fractional effect of start/stop timing. Repeats reveal variation but do not correct a miscount, the wrong equilibrium reference or perspective error. A short, nearly sinusoidal record does not establish constant amplitude over a much longer interval.

The motion-graph method shows how calibrated positions can also give approximate velocities and accelerations.

Phase specifies the stage of the cycle

Phase expresses the stage of an oscillation as an angle relative to a chosen reference. One cycle advances phase by 2π rad. Phase difference compares two stages; for traces at the same frequency, a positive time separation Δt between corresponding stages gives the phase-difference magnitude:

|Δφ| = 2πΔt/T

State which trace reaches a corresponding stage first. A quarter-cycle advance is π/2 rad; half a cycle is π rad. Equal displacement at one instant does not establish equal phase, because the bodies may be moving in opposite directions.

An earlier corresponding peak identifies the lead

Solid blue x1 = 0.0500 sin(4.00t) m. Dashed brown x2 = 0.0500 cos(4.00t) m. Both have period T = π/2 s.

The cosine displacement leads the sine displacement by one quarter cycleTwo displacement curves share axes in metres and seconds, amplitude 0.0500 metres and period pi over two seconds. The solid sine trace x1 crosses zero with positive slope at time zero and has its positive peak at T over four. The dashed cosine trace x2 is already at its positive peak at time zero. A bracket between those peak times marks T over four, so x2 leads x1 by pi over two radians. Both curves continue through the same full period. Rounded seconds sit beneath exact quarter-period labels. The graph compares displacements with the same units, not unlike motion quantities.00.000T/40.393T/20.7853T/41.178T1.571-0.0500+0.050T/4Displacement / mElapsed time t / s

x2 reaches its positive peak first, so x2 leads x1 by T/4 in time, corresponding to π/2 rad in phase. The horizontal spacing measures a time difference; the phase difference is an angle.

Both traces have the same amplitude and frequency. The second reaches its corresponding positive peak a quarter-cycle earlier, so it leads the first by π/2 rad.

For x1 = x0 sin(ωt) and x2 = x0 sin(ωt + π/2), x2 leads x1 by T/4. At t = 0, x2 is at its positive extreme while x1 crosses equilibrium towards positive x.