Conditions and equations of simple harmonic motion
Oscillation and equilibrium — An oscillation is repeated motion about an equilibrium position. Displacement x is measured from that equilibrium and can be signed, while amplitude A is the maximum displacement magnitude. A periodic motion need not be simple harmonic: the force-displacement relationship decides whether the SHM model applies.
Restoring condition — Simple harmonic motion requires acceleration proportional to displacement and opposite in direction, a = −ω²x. For constant mass this corresponds to resultant restoring force F = −kx with ω² = k/m. A force pointing towards equilibrium is insufficient if its magnitude is not proportional to displacement; other force forms can give different oscillations.
Meaning of angular frequency — Angular frequency ω = 2πf = 2π/T is in rad s⁻¹. It sets the phase rate, not a literal rotation of the oscillating object. Larger ω means a larger acceleration magnitude at the same displacement, and shorter period. Do not substitute f in hertz for ω without the 2π conversion.
Displacement solution — For an undamped oscillator released from rest at x = +A at t = 0, x = A cosωt. Different starting conditions require a phase shift or an equivalent sine expression. The amplitude is fixed in this ideal free model; cosωt is dimensionless and ωt is interpreted in radians.
Velocity and acceleration — Differentiating x gives v = −Aω sinωt and a = −Aω² cosωt = −ω²x. Maximum speed magnitude Aω occurs at equilibrium; maximum acceleration magnitude Aω² occurs at the extremes. At an extreme velocity is zero but acceleration is largest, so zero velocity does not imply no force.
Phase relationships — Velocity is a quarter cycle out of phase with displacement for sinusoidal SHM; acceleration is in antiphase with displacement. For the chosen +A release, velocity becomes negative immediately as the body moves towards equilibrium. Verify signs on a common time axis rather than memorise a lead/lag phrase without its convention.The three key positions distinguish zero velocity at extremes from zero acceleration at equilibrium. At equilibrium the speed is maximal but the sign of velocity depends on the crossing direction. Acceleration always opposes nonzero displacement.
Speed at a position — Eliminate phase using sin² + cos² = 1 to obtain v² = ω²(A² − x²), so speed is ω√(A² − x²). The signed velocity has two possible signs at most positions, depending on which way the body is moving. At x = ±A speed is zero and at x = 0 it is maximum.
Worked SHM model — For A = 0.040 m and f = 2.0 Hz, ω = 4π rad s⁻¹. Maximum speed is 0.040 × 4π = 0.503 m s⁻¹ and maximum acceleration is 0.040(4π)² = 6.32 m s⁻². At x = +0.020 m, acceleration is −3.16 m s⁻²; speed magnitude is about 0.435 m s⁻¹ and needs a direction to give signed velocity.
Springs pendulums and graph evidence
Mass-spring derivation — For a light ideal spring operating in its linear range, restoring resultant −kx gives m d²x/dt² = −kx. Comparing with SHM yields ω = √(k/m) and T = 2π√(m/k). The spring's own mass and damping are neglected in this simplest result; both can matter in accurate measurements.
Vertical equilibrium shift — For a vertical spring with downward extension y, equilibrium has ky₀ = mg. Writing displacement x = y − y₀ gives resultant mg − k(y₀ + x) = −kx. Gravity shifts equilibrium but cancels from the displacement equation, so the ideal small-oscillation period remains 2π√(m/k).
Pendulum derivation — A simple pendulum has restoring tangential force −mg sinθ. For small angles in radians, sinθ ≈ θ and arc displacement x = lθ, giving a ≈ −(g/l)x. Thus T ≈ 2π√(l/g), independent of bob mass under the model. At large amplitude the approximation fails and the period is not strictly amplitude independent.
Pendulum length and assumptions — Measure l from pivot to bob centre of mass, not just string length. Assume a small dense bob, light inextensible string, one whose mass is negligible and whose length is treated as constant, or suitable equivalent, low friction and small angle. Air resistance, pivot friction, finite amplitude and uncertain effective length can cause departures; changing bob mass can also change these nonideal effects even though ideal mass cancels.
Using oscillation periods — For a pendulum plot T² against l to infer gradient 4π²/g; for a spring plot T² against attached mass to infer 4π²/k, allowing an intercept where the spring's effective mass contributes. Time many complete cycles and divide by their number, using a consistent reference crossing and direction to avoid half-cycle miscounts.
Displacement-time graph — A tangent gradient gives instantaneous velocity. At maxima and minima the slope is zero; at equilibrium crossings slope magnitude is greatest. An envelope joins or outlines successive amplitude peaks. A decaying envelope indicates damping, so one constant-amplitude cosine no longer describes the full record. Read a complete cycle between equivalent positions with the same direction of motion.
Velocity-time graph — The gradient gives acceleration and signed area over an interval gives displacement change. For SHM, velocity extrema coincide with displacement zero and acceleration zero. A full-cycle signed velocity area is zero while distance travelled in one complete undamped cycle is 4A, not zero.
Acceleration-displacement test — Plot a against x: ideal SHM gives a straight line through equilibrium with negative slope −ω². A sinusoidal-looking trace over a short window alone is weaker evidence for this force condition. A positive a-x slope indicates acceleration away from the chosen equilibrium rather than stable SHM.The acceleration-displacement line crosses the equilibrium origin and has negative slope minus angular-frequency squared. The vertical and horizontal zero references are shown at the centre so both signs are visible. The picture is an ideal model rather than numerical measured data.
Energy and damping
Energy in ideal SHM — For a mass-spring system with displacement from equilibrium, oscillation potential energy is ½kx² and kinetic energy is ½mv². Total oscillation energy is ½kA² = ½mω²A², fixed for the ideal undamped free motion. At extremes it is all potential and at equilibrium all kinetic within this model.
Vertical-system energy — For a vertical spring, actual elastic extension is y₀ + x. The combined spring and gravitational potential changes simplify to ½kx² plus a reference constant. Do not identify ½kx² with the spring's actual absolute elastic energy alone; it is the effective oscillation potential relative to equilibrium.
Energy example — A spring of k = 100 N m⁻¹ oscillates with amplitude 0.050 m. Total oscillation energy is ½ × 100 × 0.050² = 0.125 J. At x = 0.030 m, potential is 0.045 J and kinetic is 0.080 J. Doubling amplitude gives four times the oscillation energy while period remains unchanged in the ideal linear model.
Damping mechanism — Damping transfers oscillation energy to surroundings or internal modes through resistive effects such as air drag or internal friction. Free oscillation amplitude decreases with time; total energy of the full system remains conserved. An exponential amplitude envelope is a useful model for some damping laws, not a universal shape for every friction mechanism.
Dissipation and amplitude — Because oscillation energy is proportional to amplitude squared, a reduction of amplitude to half corresponds to energy falling to one quarter in the same linear oscillator. A plotted amplitude decline does not equal the same fractional energy decline. Use actual measured envelopes and state the model when estimating energy losses per cycle.
Plastic deformation — A ductile component can dissipate mechanical energy through permanent deformation, reducing subsequent oscillation or impact motion. This differs from ideal reversible spring storage: the original equilibrium shape may change and F = −kx may no longer hold. Evaluate the material and loading history before continuing to use the same SHM equation after yielding.
Forced oscillations resonance and unknown mass
Free and forced motion — A free oscillator moves after an initial disturbance under its own restoring and damping interactions. A forced oscillator receives a continuing periodic drive and, after the initial temporary motion (the transient response) has died away, responds at the drive frequency. Its natural frequency is the characteristic free-response frequency, not necessarily equal to every imposed frequency.
Resonance and phase — For weak damping, steady-state amplitude becomes large when drive frequency is near the natural frequency because energy transfer per cycle can efficiently replace losses. In the ideal undamped resonant model amplitude can grow without a finite steady limit; real systems have damping and nonlinear or safety limits. The response depends on phase as well as frequency.
Resonance curves — Plot steady-state amplitude against drive frequency for constant driving strength and comparable damping. More damping produces a lower, broader peak; for displacement response it can also shift the peak below the undamped natural frequency. Do not claim every damped system peaks at exactly the same f₀ or use peak height comparisons when drive amplitude also changed.Qualitative steady displacement-response curves at the same drive strength contrast weak and stronger damping. The stronger-damped peak is lower, broader and slightly shifted; no universal quantitative curve shape is asserted.
Applications and evaluation — Resonance can support musical sound or sensitive measurements, yet unwanted driven motion can damage machinery or structures. Added damping lowers the response but dissipates energy and can affect performance. Identify the actual drive, restoring mechanism, frequency range and measured evidence instead of treating every vibration as resonance.
CP16 calibration design — Use the approved driven oscillator and a set of known attached masses. For each mass, sweep drive frequency around the relevant resonance with controlled drive amplitude, wait for a steady response and identify the peak consistently. Keep the spring or restoring arrangement and geometry unchanged and prevent masses falling or excessive excursions.
CP16 mass graph — For an approximately ideal mass-spring oscillator with weak damping, T² = (4π²/k)(m + meffective), so plotting 1/fres² against known attached mass gives an approximately straight calibration with possible nonzero intercept. The intercept can account for effective apparatus mass rather than being forced to zero. This model needs validation for the chosen driven system.A schematic straight calibration of inverse resonant-frequency squared against attached mass includes a possible effective-apparatus-mass intercept. An unknown is inferred using the tested arrangement and fit uncertainty rather than forcing zero intercept.
Unknown mass inference — An illustrative calibration gives 1/fres² = (0.40 s² kg⁻¹)m + 0.020 s². For an unknown with fres = 2.0 Hz, 1/fres² = 0.250 s² and m = (0.250 − 0.020)/0.40 = 0.575 kg. Include interpolation and fit uncertainty; a mass outside the calibrated range is an extrapolation.
CP16 uncertainty and limits — A broad resonance peak makes frequency selection uncertain; repeat sweeps in both directions and estimate a frequency range. The fractional contribution from f to 1/f² is approximately twice Δf/f. Maintain drive and damping between calibration and unknown runs, avoid spring yielding, and explain how altered apparatus conditions can bias the inferred mass.
Test yourself
50 questions · Random sets of 10. These quick checks support revision; practise longer explanations and justified judgements too.
Mr Kidman's Classroom
