Simple Harmonic Motion

Restoring forces and sinusoidal motion

Simple harmonic motion arises when the net force near a stable equilibrium is proportional to displacement from that equilibrium and points back toward it. Let $x$ denote displacement from the equilibrium position. The restoring force is

F_{x} = - k x,

where $k > 0$ is the stiffness. The sign is part of the definition. A positive displacement produces a negative force; a negative displacement produces a positive force. The force vanishes at equilibrium and grows linearly with distance from it.

Newton's second law for a mass $m$ gives

m \overset{x}{¨} = - k x, \overset{x}{¨} + ω_{0}^{2} x = 0, ω_{0} = \frac{k}{m} .

The quantity $ω_{0}$ is the natural angular frequency in $rad s^{- 1}$ . Its associated period and frequency are

T = \frac{2 π}{ω _{0}} = 2 π \frac{m}{k}, f = \frac{1}{T} = \frac{1}{2 π} \frac{k}{m} .

The mass and stiffness set the period. Amplitude has no effect in the ideal linear model. A measured amplitude-dependent period indicates finite geometry, nonlinear stiffness, a changing support condition, or a measurement artifact.

A horizontal spring--mass oscillator measured from equilibrium. The origin marks where the spring force vanishes; the mass is drawn displaced a distance

x

to the right, so the restoring force points back toward the origin, and an equal displacement to the left would give an equal, opposite force.

On a force--displacement graph, the slope is $- k$ and the area under the curve is the work. From $x_{i}$ to $x_{f}$ ,

W_{s} = x_{i} \int x_{f} (- k x) d x = - \frac{1}{2} k (x_{f}^{2} - x_{i}^{2}) .

The work is negative while a mass moves farther from equilibrium and positive while the spring pulls it inward. This sign pattern explains the exchange between kinetic and spring potential energy without requiring a separate force diagram at every position.

Restoring force versus displacement for a linear spring. The line passes through the origin with negative slope

- k

; a positive displacement

x

produces a negative force and a negative displacement a positive force, so the force always points back toward equilibrium.

A stable equilibrium requires positive curvature of potential energy. Near an equilibrium coordinate $q_{0}$ , expand a smooth potential $U (q)$ :

U (q) \approx U (q_{0}) + \frac{d U}{d q}_{q_{0}} (q - q_{0}) + \frac{1}{2} \frac{d ^{2} U}{d q ^{2}}_{q_{0}} (q - q_{0})^{2} .

At equilibrium the first derivative is zero. If the second derivative is positive, the local form is

U (q) \approx U (q_{0}) + \frac{1}{2} k (q - q_{0})^{2}, k = \frac{d ^{2} U}{d q ^{2}}_{q_{0}} .

Thus harmonic motion is the universal small-displacement approximation around a stable minimum. The coordinate can be a translation, an angle, a twist, a fluid surface displacement, or a normal-mode coordinate. The approximation is assessed by the range over which the measured force graph remains linear.

Local potential shapes near equilibria. A positive-curvature minimum (left) gives a restoring force and a harmonic approximation; a negative-curvature maximum (right) gives an accelerating departure. The curvature at the minimum sets the effective stiffness used in the small-oscillation equation.

A vertical spring has a nonzero static extension under gravity. Let $y$ measure distance downward from the support and let $y_{eq} = m g / k$ . Defining $x = y - y_{eq}$ removes the constant gravitational force:

m \overset{y}{¨} = m g - k y ⟹ m \overset{x}{¨} + k x = 0.

Gravity shifts the equilibrium position; it does not change the small-oscillation frequency for an ideal vertical spring. Measuring displacement from the unstretched length while inserting $x$ into the horizontal-spring equation produces an incorrect constant-force term.

Sinusoidal solution and initial conditions.

The general solution of the simple-harmonic equation is

x (t) = A cos (ω_{0} t + ϕ),

where $A \geq 0$ is amplitude and $ϕ$ is the phase constant. A sine form is equally valid. The choice should be driven by the measured initial state, not by a memorized preference. Differentiation gives

v (t) = \overset{x}{˙} = - ω_{0} A sin (ω_{0} t + ϕ),

a (t) = \overset{x}{¨} = - ω_{0}^{2} A cos (ω_{0} t + ϕ) = - ω_{0}^{2} x (t) .

Acceleration is opposite displacement and reaches its largest magnitude at the turning points. Speed is largest at equilibrium. The phase relation is fixed: the velocity curve is shifted by one quarter period relative to displacement, and acceleration is shifted by one half period.

Displacement, velocity, and acceleration through more than one harmonic cycle. Velocity crosses zero where displacement is extremal, acceleration is the mirror of displacement, and the three curves keep their phase separations regardless of amplitude, mass, or stiffness in the ideal linear model.

Initial displacement $x_{0}$ and initial velocity $v_{0}$ provide a form that avoids manual quadrant decisions:

x (t) = x_{0} cos (ω_{0} t) + \frac{v _{0}}{ω _{0}} sin (ω_{0} t) .

The amplitude follows from the initial state,

A = x_{0}^{2} + (\frac{v _{0}}{ω _{0}})^{2},

and the phase can be obtained from a quadrant-aware angle calculation,

ϕ = atan2 (- v_{0} / ω_{0}, x_{0}) .

The $atan2$ argument order matches the conventional pair $(sin ϕ, cos ϕ)$ . A one-argument inverse tangent discards the quadrant and can return a phase differing by $π$ , which reverses the initial displacement and velocity signs.

Initial-state geometry for a harmonic oscillator. The horizontal leg is the initial displacement

x_{0}

and the vertical leg is the initial velocity scaled by

ω_{0}

; the hypotenuse gives the amplitude

A

, while the oriented angle gives the phase constant

ϕ

with its correct quadrant.

Energy, phase, and equivalent stiffness

The total mechanical energy of an ideal spring oscillator is the sum of kinetic and spring potential energy:

E = \frac{1}{2} m \overset{x}{˙}^{2} + \frac{1}{2} k x^{2} .

Differentiate this expression and substitute $m \overset{x}{¨} = - k x$ :

\frac{d E}{d t} = m \overset{x}{˙} \overset{x}{¨} + k x \overset{x}{˙} = \overset{x}{˙} (m \overset{x}{¨} + k x) = 0.

Energy conservation supports both the derivation and a measurement check. At a turning point $x = \pm A$ , velocity is zero and

E = \frac{1}{2} k A^{2} .

At equilibrium $x = 0$ , spring potential energy is zero and

E = \frac{1}{2} m v_{max}^{2} .

Equating the two gives

v_{max} = ω_{0} A .

The amplitude has dimensions of length; the maximum speed has dimensions of length per time. A data table containing both quantities should satisfy this relation within the uncertainties of displacement scale, timing, and the linear-spring assumption.

Energy exchange in a spring oscillator against position. At either turning point all mechanical energy is spring potential energy; at the equilibrium crossing all of it is kinetic; intermediate positions split between the two forms while their sum stays on the constant total-energy line.

The position dependence of speed follows directly:

v (x) = \pm ω_{0} A^{2} - x^{2} .

The sign selects the direction of travel. A positive square root at a given displacement represents motion toward increasing $x$ ; a negative square root represents motion toward decreasing $x$ . Squaring the relation avoids the sign choice:

v^{2} = ω_{0}^{2} (A^{2} - x^{2}) .

A plot of $v^{2}$ against $x^{2}$ is a straight line with intercept $ω_{0}^{2} A^{2}$ and slope $- ω_{0}^{2}$ . This reduction applies when a velocity estimate is available from a motion sensor or a fitted derivative of the displacement signal. It also identifies departures from energy conservation: systematic curvature can result from a nonlinear spring or from energy loss.

The potential-energy curve contains the allowed range of motion. A horizontal line at the total energy intersects the parabola $U (x) = \frac{1}{2} k x^{2}$ at $x = - A$ and $x = A$ . Motion outside that interval would require negative kinetic energy and is therefore excluded by the model. If a sensor trace crosses an inferred turning point, check the energy reference, the amplitude extraction, and the calibration before assigning physical meaning to the excess.

Energy methods solve many amplitude questions without finding phase. A mass released from rest at $x_{0}$ has $A = ∣ x_{0} ∣$ . A mass launched at equilibrium with speed $v_{0}$ has $A = ∣ v_{0} ∣/ ω_{0}$ . A general initial state has

\frac{1}{2} k x_{0}^{2} + \frac{1}{2} m v_{0}^{2} = \frac{1}{2} k A^{2},

which reproduces the initial-state amplitude formula. The calculation assumes that the spring is measured from equilibrium and that no external work acts after the initial state is set.

A spring's work can be compared with force-sensor data by integrating the measured force curve. The stored energy is the negative of work done by the spring in moving from a turning point to equilibrium. A hysteresis loop in a loading--unloading graph has nonzero enclosed area and signals mechanical energy lost per cycle. That behavior belongs to a damped or viscoelastic model, even if the displacement trace appears nearly sinusoidal over a short interval.

The phase-space equation follows by dividing the energy equation by $k$ :

x^{2} + (\frac{v}{ω _{0}})^{2} = A^{2} .

A scaled phase portrait is a circle of radius $A$ . In unscaled coordinates it is an ellipse. The direction around the curve follows the sign of velocity. Starting at $x = A$ with zero speed, the state initially travels downward in the $(x, v / ω_{0})$ plane because acceleration is negative. Starting at $x = - A$ , it initially travels upward. These directional checks link the algebraic phase constant to a physical release state.

Scaled phase portrait for simple harmonic motion in the

(x, v / ω_{0})

plane. The path is a circle of radius equal to the amplitude; left and right points are turning points with zero speed, while top and bottom points are equilibrium crossings with maximum speed.

The phase-space radius also makes energy scaling transparent. Doubling amplitude quadruples total energy. Doubling stiffness at fixed amplitude doubles energy. Doubling mass at fixed amplitude and frequency changes kinetic energy, but a self-consistent simple-harmonic oscillator changes its frequency through $ω_{0}^{2} = k / m$ . Comparisons must state which physical quantities are held fixed; otherwise an apparent energy contradiction can arise from changing several parameters at once.

Equivalent stiffness and physical linear oscillators.

A spring assembly has an effective stiffness only after its coordinate and loading condition are specified. For springs in parallel, the displacement of each spring is the same and forces add:

F = F_{1} + F_{2} = (k_{1} + k_{2}) x, k_{∥} = k_{1} + k_{2} .

For springs in series, the force through each spring is the same and extensions add:

x = x_{1} + x_{2} = F (\frac{1}{k _{1}} + \frac{1}{k _{2}}), \frac{1}{k _{series}} = \frac{1}{k _{1}} + \frac{1}{k _{2}} .

The softer element controls a series combination. If $k_{2} ≫ k_{1}$ , then $k_{series}$ is close to $k_{1}$ , and most extension occurs in the softer spring. A parallel combination becomes stiffer than either individual spring because both contribute force for the same displacement.

Parallel and series spring reductions. Parallel springs share one displacement and their forces add, giving

k_{∥} = k_{1} + k_{2}

; series springs carry one force and their extensions add, giving

1/ k_{series} = 1/ k_{1} + 1/ k_{2}

. Each drawing fixes the coordinate before either effective-stiffness formula applies.

The formulas assume massless connectors and one-dimensional motion. A connector with substantial mass introduces another coordinate. A spring mounted at an angle changes the relationship between spring extension and the chosen laboratory coordinate. A spring whose support moves relative to the frame changes the reference position. These effects are often small, but their omission must be justified by a geometry or mass-ratio estimate.

A static calibration applies several known forces and records the resulting equilibrium displacements. The slope of $F$ versus $x$ yields the effective stiffness. Repeating the loading in both directions tests hysteresis. The dynamic frequency prediction

f_{pred} = \frac{1}{2 π} \frac{k _{static}}{m _{eff}}

can then be compared with the measured period. Agreement links static force response to dynamic inertia. A difference can indicate a mass that was omitted from $m_{eff}$ , nonlinear stiffness at the chosen amplitude, or a support that moves with the oscillator.

A vertical assembly permits a direct stiffness estimate from its static sag. An added mass $Δ m$ changes equilibrium extension by

Δ y_{eq} = \frac{Δ m g}{k}, k = \frac{Δ m g}{Δ y _{eq}} .

Use a difference between two loaded states rather than the total extension from an unknown unstretched length. The difference removes an initial preload and reduces sensitivity to a support-location offset. The added mass must move with the oscillator during the dynamic trial. If a mass is used only for calibration and then removed, the predicted period must use the mass present during the oscillation.

Torsional motion is another linear oscillator. A rigid body suspended by a wire or torsion rod experiences an approximately linear torque

τ = - κ θ,

where $θ$ is angular displacement and $κ$ is torsion constant with units $N m ra d^{- 1}$ . Rotational dynamics gives

I \ddot{θ} + κ θ = 0, ω_{0} = \frac{κ}{I}, T = 2 π \frac{I}{κ} .

The correspondence with a translating spring oscillator is $x \leftrightarrow θ$ , $m \leftrightarrow I$ , and $k \leftrightarrow κ$ . The coordinate substitution preserves the energy form:

E = \frac{1}{2} I \dot{θ}^{2} + \frac{1}{2} κ θ^{2} .

Angle must be expressed in radians for the torque constant and angular frequency relations to have their stated form. A degree value can be converted before substitution, but an unconverted degree value gives a numerical phase and energy error.

Torsional oscillator. Rotating the disk by an angle

θ

twists the suspension wire and produces a restoring torque

τ = - κ θ

; the rotational inertia

I

replaces mass and the torsion constant

κ

replaces linear stiffness in the harmonic-motion equation.

A physical system can contain both translation and rotation. A mass hung from a spring may rotate slightly; a rolling object on a spring has translational and rotational kinetic energy; a pendulum bob can swing while its support flexes. A single-coordinate harmonic description is valid only when other motions are negligible or remain locked to the chosen coordinate. Video tracking of more than one marker can expose a rotation that a one-dimensional displacement sensor cannot see.

Equivalent stiffness can also be inferred from the potential-energy curvature. For two springs attached to a mass in a symmetric geometry, write total spring energy as a function of the chosen displacement, expand about equilibrium, and identify the coefficient of $x^{2} /2$ . This approach avoids sign mistakes in force components and extends directly to springs mounted at angles. The physical geometry still controls the result: a spring nearly perpendicular to the motion can have a small first-order extension and a different effective stiffness than its axial constant suggests.

Circular reference motion, phase, and timing.

Uniform circular motion gives a geometric representation of harmonic motion. Let a point move around a circle of radius $A$ at constant angular speed $ω_{0}$ . If its angular coordinate is $θ = ω_{0} t + ϕ$ , the projection onto a diameter is

x = A cos θ = A cos (ω_{0} t + ϕ) .

The projection has the same period as the circular motion. Hooke's law determines the physical spring force; the circular point gives a compact way to represent phase and the fixed relationships among displacement, velocity, and acceleration. Differentiating the projection produces the same harmonic kinematics as the spring equation.

Circular reference motion and its horizontal projection. The rotating radius has fixed length

A

; its projection onto the horizontal diameter varies as

x = A cos (ω_{0} t + ϕ)

, so one revolution advances the oscillator through one full period and fixes the meaning of the phase angle.

The reference circle helps distinguish angular frequency from phase. Angular frequency controls how rapidly the radius advances. Phase specifies the radius orientation at a selected time. A change in time origin $t^{'} = t - t_{s}$ changes the phase constant:

A cos (ω_{0} t + ϕ) = A cos (ω_{0} t^{'} + ϕ^{'}), ϕ^{'} = ϕ + ω_{0} t_{s} .

Two analysts can report different phase constants for the same physical trace if they use different trigger times. Frequency, period, amplitude, and energy do not depend on that arbitrary choice. Phase difference between two synchronized traces is meaningful because the common time origin cancels.

For two harmonic records at the same angular frequency,

x_{1} = A_{1} cos (ω_{0} t + ϕ_{1}), x_{2} = A_{2} cos (ω_{0} t + ϕ_{2}),

the relative phase is

Δ ϕ = ϕ_{2} - ϕ_{1} .

A time separation $Δ t$ between equivalent peaks gives

Δ ϕ = ω_{0} Δ t = 2 π \frac{Δ t}{T} .

Use corresponding features: two positive peaks, two negative peaks, or two same-direction equilibrium crossings. Comparing a positive peak with a negative peak adds a half-cycle offset and reverses the intended conclusion.

Phase measurements need a stated convention for the direction of increasing angle. The cosine form used here places a positive displacement maximum at phase zero. The sine form places a positive zero crossing with positive velocity at phase zero. Both conventions are mathematically consistent. A phase label from an instrument should state whether it is referenced to a trigger edge, a displacement channel, a force channel, or a synthesized signal.

Sampling imposes a timing resolution. A record with sample interval $Δ t_{s}$ has a phase increment $ω_{0} Δ t_{s}$ between adjacent samples. Peak interpolation can improve sub-sample timing when the waveform is smooth and noise is small. Directly selecting the largest sampled value biases a peak-time estimate toward the sampling grid. A full sinusoidal fit uses all retained samples and usually provides a more stable phase estimate, but the fit residual must remain consistent with a single-frequency waveform.

The circular representation also gives an acceleration construction. The reference-point acceleration has magnitude $ω_{0}^{2} A$ and points toward the circle center. Its horizontal projection is $- ω_{0}^{2} x$ , matching the spring equation. The relation relies on constant angular speed. A nonuniform circular motion produces additional tangential acceleration and does not represent simple harmonic motion.

A phase-space record and a circular reference plot use related geometry but different axes. The reference circle places the spatial projection on one diameter and uses a fictitious second coordinate. A scaled phase portrait uses actual displacement and actual velocity divided by $ω_{0}$ . Confusing the two can reverse an inferred direction of motion or assign physical units to the reference-circle vertical axis. Use the phase portrait for measured state data and the reference circle for phase geometry.

Oscillator measurement and parameter estimation

A period measurement begins with a physical definition of one cycle. A displacement trace uses the time between successive positive peaks, successive negative peaks, or successive upward equilibrium crossings. Use the same feature throughout one analysis. Zero crossings are often sharper than broad peaks, but their timing can be biased by a drifting baseline. Peaks avoid a baseline crossing criterion but can be poorly resolved when samples are sparse. A global sinusoidal fit yields a third estimate and should be compared with a direct feature-based period.

Measure many cycles when the oscillator remains close to constant amplitude. If $t_{N}$ is the elapsed time across $N$ whole periods,

T = \frac{t _{N}}{N} .

A fixed timing uncertainty in $t_{N}$ is divided by $N$ . The interval must start and end on equivalent waveform features. Counting $N$ intervals requires attention: $N + 1$ marked peaks bound $N$ periods. A clear analysis record lists the feature indices, their timestamps, and the resulting interval count.

The standard uncertainty in a multi-cycle result has contributions from time base, feature location, cycle count, and repeatability. If the elapsed-time standard uncertainty is $σ_{t}$ and the count is exact,

σ_{T} = \frac{σ _{t}}{N} .

Repeat full releases rather than repeatedly reading the same trace. The spread between independent releases includes changes in release position, air currents, support motion, and sensor noise. A period that changes systematically with initial amplitude signals a model departure rather than random scatter. Record amplitude with every period measurement and plot $T$ against amplitude before combining runs.

Displacement calibration maps a sensor reading to physical length. A linear calibration has the form

x = a (V_{s} - V_{off}),

where $a$ is scale and $V_{off}$ is the sensor output at the chosen equilibrium. Acquire several known positions that span the oscillation range. Fit the slope and intercept, then plot residuals in units of length. A calibration curve with curvature should be used only over a limited monotonic range or replaced by a suitable nonlinear model. A linear fit forced through a curved response can create an apparent amplitude-dependent frequency through distorted turning-point positions.

A force calibration for $k$ should use the same coordinate and attachment geometry as the dynamic trial. Place known masses gently on a vertical system or pull a horizontal system through measured positions with a calibrated force sensor. The mass of a hook, force probe, and fixture belongs in the load when its weight or inertia acts on the spring. A force sensor reading includes its own zero offset and may drift with temperature. Collect a zero reading before and after the force sweep.

The dynamic estimate

k_{dyn} = m_{eff} (\frac{2 π}{T})^{2}

offers an independent check. Do not average $k_{static}$ and $k_{dyn}$ until their difference has been explained. Agreement supports the linear single-mass model. A static value larger than a dynamic value can arise when the support or spring itself contributes unmodeled inertia. A dynamic value that grows with amplitude can arise when the spring stiffens as it stretches.

Amplitude extraction needs the same care as period extraction. A centered trace uses half the difference between a positive peak and the adjacent negative peak:

A \approx \frac{x _{max} - x _{min}}{2} .

This estimate cancels a constant baseline offset. It does not cancel a baseline that drifts between the two extrema. A global model with an explicit baseline can be preferable for a long trace. Report whether amplitude comes from a single pair of peaks, an average over several cycles, or a fitted coefficient.

Energy measurements amplify amplitude uncertainty because $E$ is proportional to $A^{2}$ . Fixed stiffness gives

\frac{σ _{E}}{E} \approx \frac{σ _{k}}{k} + 2 \frac{σ _{A}}{A}

as a conservative first-order magnitude estimate when parameter covariance is neglected. A ten-percent amplitude uncertainty therefore contributes about twenty percent relative energy uncertainty. Energy comparisons between two amplitudes are often more reliable when the same sensor scale cancels in the ratio, but a changing baseline can still bias the smaller amplitude strongly.

A measured acceleration channel is another consistency test. A harmonic trace has an acceleration-versus-displacement plot with slope $- ω_{0}^{2}$ . Obtain acceleration from a calibrated accelerometer or from a smoothed analytic derivative of a displacement fit. Raw numerical second differences magnify high-frequency sensor noise, so a jagged acceleration graph does not by itself disprove the harmonic model.

A frequency-amplitude test directly checks the model. Record several releases with different measured amplitudes and calculate period using the same multi-cycle procedure. The ideal prediction is a horizontal $T (A)$ graph. A statistically significant slope can be fit and reported. Its physical interpretation depends on the apparatus: a stretched spring can stiffen, an angled spring can change geometric projection, and a large-angle pendulum has a longer period than its small-angle limit. The graph identifies the range in which a single harmonic frequency is defensible.

Period against release amplitude. A horizontal trend supports the amplitude-independent period of a linear oscillator; a resolved upward slope at larger amplitude marks the range where an effective stiffness or a nonlinear model must replace one constant natural frequency.

A reproducible measurement record contains raw sensor values, calibration points, timestamps, peak or fit settings, environmental conditions, and the selected data window. Graph axes need units and a stated reference. A reader should be able to recompute $T$ , $A$ , $k$ , and $ω_{0}$ from the stored measurements. A screen image without timing metadata or sensor scale supports qualitative discussion but cannot establish a numerical natural frequency.

Parameter estimation from complete traces.

A full displacement record can be modeled as

x_{model} (t) = x_{c} + A cos [ω_{0} (t - t_{c}) + ϕ],

where $x_{c}$ is equilibrium position and $t_{c}$ is the chosen reference time. Fitting all samples uses more information than reading a single peak. The residual

r_{i} = x_{i} - x_{model} (t_{i})

should be plotted against time and displacement. A residual that alternates in sign with the oscillation can indicate a slightly wrong frequency. A residual that grows with displacement can indicate nonlinear stiffness or sensor saturation. A slow residual trend can indicate baseline motion. These diagnoses require the residual shape together with its root-mean-square value.

Worked reductions and model validity

A spring oscillator has mass $m = 0.250 kg$ and measured stiffness $k = 40.0 N m^{- 1}$ . Its natural angular frequency and period are

ω_{0} = \frac{40.0}{0.250} = 12.65 rad s^{- 1},

T = \frac{2 π}{12.65} = 0.4967 s .

Suppose the measurement begins with $x_{0} = 0.0800 m$ and $v_{0} = - 0.300 m s^{- 1}$ . The amplitude is

A = (0.0800 m)^{2} + (\frac{- 0.300 m s ^{- 1}}{12.65 rad s ^{- 1}})^{2} = 0.0834 m .

The negative initial velocity means that the mass moves toward decreasing displacement at the selected initial time. The phase follows from the two-component state:

ϕ = atan2 (0.0237 m, 0.0800 m) = 0.288 rad .

The resulting equation is

x (t) = (0.0834 m) cos [(12.65 rad s^{- 1}) t + 0.288] .

Substitution at $t = 0$ returns $x_{0} = 0.0800 m$ . Differentiation returns $v_{0} = - 0.300 m s^{- 1}$ . These two direct substitutions are more reliable than an informal visual phase check.

The total energy can be calculated two ways:

E = \frac{1}{2} k A^{2} = \frac{1}{2} (40.0) (0.0834)^{2} = 0.139 J,

E = \frac{1}{2} k x_{0}^{2} + \frac{1}{2} m v_{0}^{2} = 0.128 J + 0.0113 J = 0.139 J .

The agreement verifies the amplitude reduction. The maximum speed and acceleration are

v_{max} = ω_{0} A = 1.06 m s^{- 1}, a_{max} = ω_{0}^{2} A = 13.3 m s^{- 2} .

The initial speed is below the maximum because the initial position is already close to a turning point. A calculation that gives an initial speed larger than $ω_{0} A$ violates the energy relation and requires a review of units, phase, or stiffness.

Series and parallel systems offer a second worked comparison. Take $k_{1} = 30.0 N m^{- 1}$ , $k_{2} = 60.0 N m^{- 1}$ , and $m = 0.500 kg$ . For the series connection,

k_{series} = (\frac{1}{30.0} + \frac{1}{60.0})^{- 1} = 20.0 N m^{- 1},

T_{series} = 2 π \frac{0.500}{20.0} = 0.993 s .

For the parallel connection,

k_{∥} = 30.0 + 60.0 = 90.0 N m^{- 1},

T_{∥} = 2 π \frac{0.500}{90.0} = 0.468 s .

The order of the periods agrees with physical reasoning. The series system has the smallest effective stiffness and longest period. The parallel system has the largest effective stiffness and shortest period. A reversed ordering usually signals that the series and parallel formulas were exchanged.

Under a static force $F = 3.00 N$ , the series system extends

x_{series} = \frac{F}{k _{series}} = 0.150 m .

Its individual extensions are $x_{1} = F / k_{1} = 0.100 m$ and $x_{2} = F / k_{2} = 0.0500 m$ , which add to the total. The parallel system extends only $0.0333 m$ . The force splits as $F_{1} = k_{1} x = 1.00 N$ and $F_{2} = k_{2} x = 2.00 N$ , which add to the applied force. These static checks validate the dynamic reduction before a period measurement is made.

The number of significant figures should follow the least precise input. A period measured from an ordinary handheld stopwatch may justify only two or three significant figures, even when a calculator produces more. Retain extra digits during intermediate calculations and round the displayed result at the end. State the measurement uncertainty separately from rounding. Rounding is a formatting choice; uncertainty is a property of the data and model.

A dimensional check helps detect formula misuse:

[\frac{m}{k}] = \frac{kg}{N m ^{- 1}} = s^{2} = s .

For torsional motion,

[\frac{I}{κ}] = \frac{kg m ^{2}}{N m ra d ^{- 1}} = s,

because radians are dimensionless in SI. The units check cannot establish the sign of a phase or the correct spring topology, but it catches a misplaced reciprocal or a stiffness entered with incompatible units.

A calculation should finish with a physical comparison. For the worked $0.500 kg$ series system, an approximately one-second period is consistent with its slow, soft response. For the parallel system, a period under half a second is consistent with its much larger stiffness. If a computed value demands motion faster than the sensor sample rate or beyond the spring's safe extension, modify the experiment before collecting data.

Validity range, coordinate choice, and model checks.

The harmonic equation is a local approximation. A general conservative coordinate $q$ requires expansion of the potential about a stable equilibrium $q_{0}$ :

U (q) = U (q_{0}) + \frac{1}{2} k (q - q_{0})^{2} + \frac{1}{6} c_{3} (q - q_{0})^{3} + \frac{1}{24} c_{4} (q - q_{0})^{4} + \dots .

The quadratic term gives simple harmonic motion. Cubic and quartic terms alter the restoring force at larger displacement:

F (q) = - k (q - q_{0}) - \frac{1}{2} c_{3} (q - q_{0})^{2} - \frac{1}{6} c_{4} (q - q_{0})^{3} - \dots .

The coefficients have physical dimensions that depend on coordinate choice. A nonzero cubic potential term produces an asymmetric force curve about the selected equilibrium. A quartic potential term can stiffen or soften the motion at large amplitude. A measured period that varies with amplitude is one observable consequence, but its sign and size depend on the full force law.

Linear range and nonlinear restoring response. The straight line is the tangent whose slope

- k

is the stiffness used in the harmonic equation; the curved force law departs from it beyond the marked interval, shifting the measured period and producing asymmetric turning points at larger displacement.

The operating range should be established from data. Collect force-displacement points across the intended displacement interval, fit the linear part, and inspect residuals. Then collect periods over several amplitudes within and beyond that interval when safe. A range can be stated operationally: for example, “the linear model was used for amplitudes below $15 mm$ , where force residuals were within the sensor uncertainty and period change was unresolved.” The statement connects the approximation to evidence rather than to a vague claim of “small” motion.

Coordinate choice determines which terms appear simple. A vertical spring coordinate measured from its static equilibrium removes gravity. A torsional coordinate measured from its untwisted equilibrium removes constant torque offsets. A bead constrained to a curved track is described most cleanly by arc length or an angle. A coordinate with nonlinear geometry can make a physically simple device look nonlinear through the transformation alone. State the coordinate before assigning $k$ , amplitude, or energy.

A uniform spring has distributed mass. If its mass $m_{s}$ is appreciable compared with the attached mass, part of the spring moves at each instant. A common approximation for a spring fixed at one end and attached to a moving mass is

m_{eff} \approx m + \frac{1}{3} m_{s} .

The factor depends on the assumed mode shape and boundary conditions. It is not a universal correction for every spring. Measure the frequency with several added masses and plot $T^{2}$ against added mass. The slope gives stiffness, while the intercept estimates the moving mass associated with the apparatus.

T^{2} = \frac{4 π ^{2}}{k} (m_{added} + m_{eff, 0}) .

This mass-loading method separates a spring's moving mass from the external mass without requiring an exact material-density model.

An effective mass can also include a sensor flag, a force probe, a hanger, or a platform. Each item belongs in the moving mass only if it moves with the selected coordinate. A cable can add stiffness and damping. An optical target can add negligible mass but alter the apparent coordinate if it is mounted away from the moving body. List the configuration for each measured period. Swapping a sensor bracket between trials can change both inertia and spring geometry.

A single-coordinate record omits additional motion. A support can vibrate, a spring can swing sideways, or a torsion disk can wobble. The resulting trace may contain a primary frequency with a small second component. Inspect a long record and a spectrum or residual plot. If two frequencies are resolved, a one-coordinate model cannot assign one exact $k$ and one exact $m$ to the complete motion. Retain the dominant-mode result only with a stated bandwidth and amplitude range. Coupled-mode analysis belongs in a separate treatment because its state requires more than position and velocity of one coordinate.

The energy relation can diagnose unmodeled external work. An ideal oscillator has a plot of

E_{i} = \frac{1}{2} m v_{i}^{2} + \frac{1}{2} k x_{i}^{2}

at many samples has one constant value within measurement uncertainty. A slow decrease suggests damping. A periodic energy variation can suggest a calibration error or an external drive. A rapid energy change near one position can suggest a mechanical stop, a magnetic interaction, or sensor clipping. Energy should be computed from smoothed velocity or a fitted derivative; unfiltered numerical differentiation can dominate the kinetic term with noise.

Check dimensions before interpreting a fit. In

x (t) = A cos (ω_{0} t + ϕ),

the argument of cosine must be dimensionless. $ω_{0} t$ is dimensionless, and $ϕ$ is an angle expressed in radians. A frequency in hertz must be converted with $ω_{0} = 2 π f$ . Substituting $f$ directly into a cosine that expects angular frequency changes the period by $2 π$ . This is one of the most common numerical errors in oscillator work.

A sign check uses the measured acceleration. For $x > 0$ , a stable harmonic oscillator must have $a < 0$ . For $x < 0$ , it must have $a > 0$ . A trace that violates this relation can result from reversed sensor polarity, an origin placed away from equilibrium, or an unstable equilibrium. The data may still describe a real motion, but the stated coordinate or model must be revised.

Energy and phase checks complement each other. The energy relation bounds speed at each displacement. The phase relation bounds the ordering of extrema and zero crossings. A record can satisfy one while failing the other: for example, a calibrated displacement scale may preserve an energy-like curve while a timing delay moves velocity phase. Apply both checks before extracting an experimental $ω_{0}$ .

Reporting and uncertainty

A numerical report should specify the model first:

m \overset{x}{¨} + k x = 0, x (t) = A cos (ω_{0} t + ϕ) .

Then state the coordinate origin, measured mass, stiffness source, amplitude range, period method, and uncertainty. A concise result might read: “For displacement measured from the static equilibrium, a $0.250 kg$ moving mass and $40.0 N m^{- 1}$ stiffness gave $T = (0.497 \pm 0.003) s$ over amplitudes from $10$ to $15 mm$ . The fitted waveform residuals were consistent with sensor noise.” Each number is tied to an observable and a model condition.

The same report should distinguish calibration uncertainty from model scope. A sensor scale may be known to one percent while the spring response has a three-percent amplitude dependence beyond the selected range. Quoting only the sensor precision would understate the uncertainty of predictions outside the validated interval. Separate random repeatability, calibration terms, and model-limitation terms whenever the data support that distinction.

Damping, driven response, normal modes, wave motion, and resonance extend simple harmonic motion by adding loss, external forcing, or additional coordinates. The linear conservative oscillator developed here sets the reference frequency, energy scale, and phase convention used by those later analyses.

Acceptance checks and uncertainty propagation.

A harmonic result can be checked without repeating the full derivation. Start with the measured period, mass, and stiffness. The three quantities must satisfy

T^{2} = \frac{4 π ^{2} m}{k} .

Plotting $T^{2}$ against $m$ for several added masses gives a straight-line test of the model. The slope is $4 π^{2} / k$ . The extrapolated mass intercept identifies moving hardware that was omitted from a nominal mass list. A curved trend can arise from nonlinear stiffness, a changing spring geometry, or a support whose response changes with load. The graph often diagnoses the apparatus more effectively than one period measurement.

The dynamic stiffness inferred from $m$ and $T$ is

k_{dyn} = \frac{4 π ^{2} m}{T ^{2}},

a first-order relative uncertainty estimate is

\frac{σ _{k}}{k} \approx \frac{σ _{m}}{m} + 2 \frac{σ _{T}}{T} .

These terms are magnitude contributions and should be replaced by a covariance calculation when mass and period estimates share a calibration or a fitted parameter. The factor of two makes timing precision important. If the period is obtained across twenty cycles, the interval count and clock uncertainty should be recorded alongside the final $T$ .

Amplitude and phase uncertainties enter different predictions. Amplitude controls energy and maximum speed:

E = \frac{1}{2} k A^{2}, v_{max} = ω_{0} A .

Phase controls the predicted state at a specified time but does not change total energy. A trigger-time uncertainty $σ_{t}$ becomes phase uncertainty

σ_{ϕ} \approx ω_{0} σ_{t} .

At high frequency, a small timestamp error can produce a substantial phase error while period remains accurately measured. State whether phase is needed for the question. A phase estimate can be omitted from an energy-only calculation; adding an arbitrary phase number creates an appearance of precision without relevant information.

A model-validation sequence can be organized around four comparisons:

Force comparison. Static force versus displacement remains linear over the amplitude range used dynamically.
Timing comparison. Period from multi-cycle feature timing agrees with period from a full-trace sinusoidal model.
Energy comparison. Energy inferred at turning points agrees with energy inferred from displacement and velocity at interior points.
State comparison. The sign and relative phase of displacement, velocity, and acceleration follow the harmonic relations.

Each comparison has a distinct failure signature. A force curve with hysteresis suggests material loss. A timing disagreement can indicate a baseline issue, sampling limit, or a waveform containing more than one frequency. An energy difference can indicate a wrong effective mass or a velocity derivative contaminated by noise. A phase error can indicate an inverted sensor channel or an unstated electronic delay.

Uncertainty budgets should separate data scatter from model limits. Repeated periods at the same amplitude quantify repeatability. Position-calibration residuals quantify sensor scale or interpolation limits. A comparison of low-amplitude and high-amplitude periods quantifies a nonlinear range effect. A known added mass has a manufacturer tolerance and a balance-measurement uncertainty. These contributions belong in different rows of a calculation because changing the instrument or changing the amplitude range addresses different sources.

A reported uncertainty has a confidence convention. State whether a value after $\pm$ is a standard uncertainty, an expanded interval with a stated coverage factor, or a practical bounds estimate. Keep units attached to every dimensional uncertainty. A phrase such as “period uncertainty is 0.003” is incomplete without seconds and a method. A phase uncertainty can be in radians or degrees, provided the unit is stated and converted consistently in equations.

Sensor bandwidth must exceed the oscillator frequency and any harmonic content needed for the selected analysis. A displacement sensor can reproduce the fundamental motion while filtering a sharp release transient. An accelerometer can have a high noise floor that dominates a double derivative. An anti-alias filter protects the sampled record but can introduce a phase delay. Characterize the measurement chain when phase, acceleration, or high-frequency residuals form part of the conclusion.

The experiment should also control external disturbances. Air currents can add variable drag to a light mass. A tabletop can transmit vibration from nearby equipment. A coil sensor can add magnetic damping when connected to a load. Thermal change can alter a polymer spring's stiffness. Measure a rest baseline before the release, repeat the trace after a delay, and compare conditions across runs. The goal is a documented range of reproducible behavior, not an isolated visually smooth trace.

A final result can be written compactly while preserving the necessary evidence:

ω_{0} = (12.65 \pm 0.08) rad s^{- 1}, T = (0.4967 \pm 0.0031) s,

with the coordinate origin, amplitude interval, and uncertainty method in the surrounding text. The two values should satisfy $ω_{0} T = 2 π$ within their combined uncertainty. If they do not, avoid reporting both as independent final measurements without explaining the difference.

The ideal simple-harmonic model sets a clear standard: one stable equilibrium, one linear restoring coefficient, one inertial parameter, and no energy transfer after the initial state. Real apparatuses approach that standard over a measured range. The force curve, period data, energy relation, phase relation, and residuals define that range. Those checks make the later use of damping, driven response, and coupled-mode models a controlled extension of a verified baseline.