---
title: Simple Harmonic Motion
module: Oscillations and Waves
moduleNumber: 7
lessonNumber: 12
order: 712
summary: >
  Any system pushed back toward equilibrium by a force proportional to its
  displacement obeys one equation, $\ddot x+\omega_0^2x=0$, and so moves sinusoidally
  at $\omega_0=\sqrt{k/m}$ whatever the amplitude. We derive that motion, follow its
  energy $E=mv^2/2+kx^2/2$ trading between kinetic and potential form at constant
  total, and read the elliptical phase-space orbit Hooke's law implies. Period,
  amplitude, velocity, and acceleration then supply redundant checks: an
  amplitude-dependent period or a curved force residual is the signature that the
  linear model has failed, and mass-loading and offset tests separate a calibration
  error from a real frequency shift.
topics: [Oscillations]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 14 — Oscillations; §§14-1–14-3"
---

## 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

$$
F_x=-kx,
$$

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

$$
m\ddot x=-kx,
\qquad
\ddot x+\omega_0^2x=0,
\qquad
\omega_0=\sqrt{\frac{k}{m}}.
$$

The quantity $\omega_0$ is the natural angular frequency in
$\mathrm{rad\,s^{-1}}$. Its associated period and frequency are

$$
T=\frac{2\pi}{\omega_0}
=2\pi\sqrt{\frac{m}{k}},
\qquad
f=\frac{1}{T}
=\frac{1}{2\pi}\sqrt{\frac{k}{m}}.
$$

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.

$$
% caption: A horizontal spring--mass oscillator measured from equilibrium. The
% origin marks where the spring force vanishes; the mass is drawn displaced a
% distance $x$ to the right, so the restoring force points back toward the origin,
% and an equal displacement to the left would give an equal, opposite force.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,thick] (0,0)--(0,2.1);
\foreach \yy in {0.2,0.55,0.9,1.25,1.6,1.95}{\draw[black] (0,\yy)--(-0.22,\yy-0.2);}
\draw[black] (0,0)--(6.6,0);
\draw[very thick] (0,1.15)--(0.45,1.15)--(0.63,1.45)--(1.0,0.85)--(1.37,1.45)--(1.74,0.85)--(2.11,1.45)--(2.48,0.85)--(2.66,1.15)--(3.3,1.15);
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\node at (3.9,1.1) {$m$};
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\node[below] at (2.9,-0.02) {origin};
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\node[acc,above] at (3.95,2.1) {restoring force};
\node[black,above] at (1.65,1.45) {spring};
\end{tikzpicture}
$$

On a force--displacement graph, the slope is $-k$ and the area under the curve is
the work. From $x_i$ to $x_f$,

$$
W_s=\int_{x_i}^{x_f}(-kx)\d x
=-\frac12k(x_f^2-x_i^2).
$$

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.

$$
% caption: Restoring force versus displacement for a linear spring. The line passes
% through the origin with negative slope $-k$; a positive displacement $x$ produces a
% negative force and a negative displacement a positive force, so the force always
% points back toward equilibrium.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,1.4)--(5.4,1.4) node[right] {displacement};
\draw[->,black] (2.6,0)--(2.6,2.9) node[above] {spring force};
\draw[acc,very thick] (0.5,2.5)--(4.7,0.35);
\draw[black,dashed] (3.7,1.4)--(3.7,0.86);
\draw[black,dashed] (2.6,0.86)--(3.7,0.86);
\node at (3.94,1.18) {$x$};
\node[acc,above] at (1.15,2.35) {slope};
\end{tikzpicture}
$$

A stable equilibrium requires positive curvature of potential energy. Near an
equilibrium coordinate $q_0$, expand a smooth potential $U(q)$:

$$
U(q)\approx U(q_0)
+\left.\frac{\d U}{\d q}\right|_{q_0}(q-q_0)
+\frac12\left.\frac{\d^2U}{\d q^2}\right|_{q_0}(q-q_0)^2.
$$

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

$$
U(q)\approx U(q_0)+\frac12k(q-q_0)^2,
\qquad
k=\left.\frac{\d^2U}{\d q^2}\right|_{q_0}.
$$

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.

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

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

$$
m\ddot y=mg-ky
\quad\Longrightarrow\quad
m\ddot x+kx=0.
$$

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

$$
x(t)=A\cos(\omega_0t+\phi),
$$

where $A\ge0$ is amplitude and $\phi$ 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

$$
v(t)=\dot x=-\omega_0A\sin(\omega_0t+\phi),
$$

$$
a(t)=\ddot x=-\omega_0^2A\cos(\omega_0t+\phi)
=-\omega_0^2x(t).
$$

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.

$$
% caption: Displacement, velocity, and acceleration through more than one harmonic
% cycle. Velocity crosses zero where displacement is extremal, acceleration is the
% mirror of displacement, and the three curves keep their phase separations
% regardless of amplitude, mass, or stiffness in the ideal linear model.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.3,1.5)--(6.1,1.5) node[right] {time};
\draw[->,black] (0.5,0.5)--(0.5,2.7) node[above] {scaled value};
\draw[acc,very thick] plot[domain=0.6:5.2,samples=200] (\x,{1.5+0.8*cos(180*(\x-0.6))});
\draw[black,very thick] plot[domain=0.6:5.2,samples=200] (\x,{1.5-0.8*sin(180*(\x-0.6))});
\draw[black,thick,densely dashed] plot[domain=0.6:5.2,samples=200] (\x,{1.5-0.8*cos(180*(\x-0.6))});
\node[acc] at (5.45,1.25) {$x$};
\node[black] at (5.45,0.74) {$v$};
\node at (5.45,1.8) {$a$};
\end{tikzpicture}
$$

Initial displacement $x_0$ and initial velocity $v_0$ provide a form that avoids
manual quadrant decisions:

$$
x(t)=x_0\cos(\omega_0t)
+\frac{v_0}{\omega_0}\sin(\omega_0t).
$$

The amplitude follows from the initial state,

$$
A=\sqrt{x_0^2+\left(\frac{v_0}{\omega_0}\right)^2},
$$

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

$$
\phi=\atanTwo(-v_0/\omega_0,x_0).
$$

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

$$
% caption: Initial-state geometry for a harmonic oscillator. The horizontal leg is
% the initial displacement $x_0$ and the vertical leg is the initial velocity scaled
% by $\omega_0$; the hypotenuse gives the amplitude $A$, while the oriented angle
% gives the phase constant $\phi$ with its correct quadrant.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.4,0.5)--(5.6,0.5) node[right] {displacement};
\draw[->,black] (0.9,0.2)--(0.9,2.7) node[above] {scaled velocity};
\draw[->,acc,very thick] (0.9,0.5)--(4.3,2.1);
\draw[black] (4.3,0.5)--(4.3,2.1);
\draw[black,thick] (1.8,0.5) arc (0:25:0.9);
\node[below] at (2.6,0.5) {$x_0$};
\node[right] at (4.3,1.25) {scaled speed};
\node[acc,above] at (2.2,1.5) {amplitude};
\node[black] at (2.0,0.78) {phase};
\end{tikzpicture}
$$


## Energy, phase, and equivalent stiffness

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

$$
E=\frac12m\dot x^2+\frac12kx^2.
$$

Differentiate this expression and substitute $m\ddot x=-kx$:

$$
\frac{\d E}{\d t}
=m\dot x\ddot x+kx\dot x
=\dot x(m\ddot x+kx)=0.
$$

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

$$
E=\frac12kA^2.
$$

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

$$
E=\frac12mv_{\max}^2.
$$

Equating the two gives

$$
v_{\max}=\omega_0A.
$$

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.

$$
% caption: Energy exchange in a spring oscillator against position. At either turning
% point all mechanical energy is spring potential energy; at the equilibrium crossing
% all of it is kinetic; intermediate positions split between the two forms while their
% sum stays on the constant total-energy line.
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\node[black] at (5.0,0.95) {kinetic};
\node[above] at (3.9,2.25) {total};
\end{tikzpicture}
$$

The position dependence of speed follows directly:

$$
v(x)=\pm\omega_0\sqrt{A^2-x^2}.
$$

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:

$$
v^2=\omega_0^2(A^2-x^2).
$$

A plot of $v^2$ against $x^2$ is a straight line with intercept
$\omega_0^2A^2$ and slope $-\omega_0^2$. 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 $U(x)=\tfrac12kx^2$ 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 $A=|v_0|/\omega_0$. A general initial state has

$$
\frac12kx_0^2+\frac12mv_0^2=\frac12kA^2,
$$

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 $k$:

$$
x^2+\left(\frac{v}{\omega_0}\right)^2=A^2.
$$

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
$(x,v/\omega_0)$ 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.

$$
% caption: Scaled phase portrait for simple harmonic motion in the
% $(x,\,v/\omega_0)$ plane. The path is a circle of radius equal to the amplitude;
% left and right points are turning points with zero speed, while top and bottom
% points are equilibrium crossings with maximum speed.
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\node[right] at (3.15,2.62) {center};
\end{tikzpicture}
$$

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
$\omega_0^2=k/m$. 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:

$$
F=F_1+F_2=(k_1+k_2)x,
\qquad
k_{\parallel}=k_1+k_2.
$$

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

$$
x=x_1+x_2
=F\left(\frac{1}{k_1}+\frac{1}{k_2}\right),
\qquad
\frac{1}{k_{\rm series}}
=\frac1{k_1}+\frac1{k_2}.
$$

The softer element controls a series combination. If $k_2\gg k_1$, then
$k_{\rm series}$ 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.

$$
% caption: Parallel and series spring reductions. Parallel springs share one
% displacement and their forces add, giving $k_\parallel=k_1+k_2$; series springs
% carry one force and their extensions add, giving $1/k_{\mathrm{series}}=1/k_1+1/k_2$.
% Each drawing fixes the coordinate before either effective-stiffness formula applies.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
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$$

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

$$
f_{\rm pred}=\frac{1}{2\pi}\sqrt{\frac{k_{\rm static}}{m_{\rm eff}}}
$$

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
$m_{\rm eff}$, 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 $\Delta m$ changes equilibrium extension by

$$
\Delta y_{\rm eq}=\frac{\Delta m\,g}{k},
\qquad
k=\frac{\Delta m\,g}{\Delta y_{\rm eq}}.
$$

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

$$
\tau=-\kappa\theta,
$$

where $\theta$ is angular displacement and $\kappa$ is torsion constant with
units $\mathrm{N\,m\,rad^{-1}}$. Rotational dynamics gives

$$
I\ddot\theta+\kappa\theta=0,
\qquad
\omega_0=\sqrt{\frac{\kappa}{I}},
\qquad
T=2\pi\sqrt{\frac{I}{\kappa}}.
$$

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

$$
E=\frac12I\dot\theta^2+\frac12\kappa\theta^2.
$$

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.

$$
% caption: Torsional oscillator. Rotating the disk by an angle $\theta$ twists the
% suspension wire and produces a restoring torque $\tau=-\kappa\theta$; the rotational
% inertia $I$ replaces mass and the torsion constant $\kappa$ replaces linear
% stiffness in the harmonic-motion equation.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,thick] (2.0,2.9)--(3.4,2.9);
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\node[right] at (3.35,1.62) {twist};
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$$

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 $\omega_0$.
If its angular coordinate is $\theta=\omega_0t+\phi$, the projection onto a
diameter is

$$
x=A\cos\theta=A\cos(\omega_0t+\phi).
$$

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.

$$
% caption: Circular reference motion and its horizontal projection. The rotating
% radius has fixed length $A$; its projection onto the horizontal diameter varies as
% $x=A\cos(\omega_0t+\phi)$, so one revolution advances the oscillator through one
% full period and fixes the meaning of the phase angle.
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\node at (2.4,2.95) {rotation};
\end{tikzpicture}
$$

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:

$$
A\cos(\omega_0t+\phi)
=A\cos(\omega_0t'+\phi'),
\qquad
\phi'=\phi+\omega_0t_s.
$$

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,

$$
x_1=A_1\cos(\omega_0t+\phi_1),
\qquad
x_2=A_2\cos(\omega_0t+\phi_2),
$$

the relative phase is

$$
\Delta\phi=\phi_2-\phi_1.
$$

A time separation $\Delta t$ between equivalent peaks gives

$$
\Delta\phi=\omega_0\Delta t
=2\pi\frac{\Delta t}{T}.
$$

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 $\Delta t_s$
has a phase increment $\omega_0\Delta t_s$ 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 $\omega_0^2A$ and points toward the
circle center. Its horizontal projection is $-\omega_0^2x$, 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 $\omega_0$. 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 $\sigma_t$ and the count is exact,

$$
\sigma_T=\frac{\sigma_t}{N}.
$$

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

$$
x=a(V_s-V_{\rm off}),
$$

where $a$ is scale and $V_{\rm off}$ 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

$$
k_{\rm dyn}=m_{\rm eff}\left(\frac{2\pi}{T}\right)^2
$$

offers an independent check. Do not average $k_{\rm static}$ and $k_{\rm dyn}$
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

$$
\frac{\sigma_E}{E}
\approx
\frac{\sigma_k}{k}
+2\frac{\sigma_A}{A}
$$

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 $-\omega_0^2$. 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.

$$
% caption: Period against release amplitude. A horizontal trend supports the
% amplitude-independent period of a linear oscillator; a resolved upward slope at
% larger amplitude marks the range where an effective stiffness or a nonlinear model
% must replace one constant natural frequency.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.5,0.6)--(5.8,0.6) node[right] {amplitude};
\draw[->,black] (0.5,0.6)--(0.5,2.8) node[above] {period};
\draw[acc,very thick] (0.8,1.55)--(5.3,1.55);
\foreach \x/\y in {1.0/1.54,1.6/1.57,2.2/1.53,2.8/1.56,3.4/1.55,4.0/1.57}{\draw[fill=white,draw=black] (\x,\y) circle (1.9pt);}
\draw[black,dashed,thick] (3.4,1.55)--(5.3,2.35);
\foreach \x/\y in {4.0/1.8,4.6/2.05,5.15/2.3}{\draw[fill=acc!10,draw=acc] (\x,\y) circle (1.9pt);}
\node[acc,below] at (2.2,1.45) {linear range};
\node[black] at (4.4,2.55) {changing period};
\end{tikzpicture}
$$

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 $\omega_0$ 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

$$
x_{\rm model}(t)
=x_c+A\cos[\omega_0(t-t_c)+\phi],
$$

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

$$
r_i=x_i-x_{\rm model}(t_i)
$$

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 $m=0.250\ \mathrm{kg}$ and measured stiffness
$k=40.0\ \mathrm{N\,m^{-1}}$. Its natural angular frequency and period are

$$
\omega_0=\sqrt{\frac{40.0}{0.250}}
=12.65\ \mathrm{rad\,s^{-1}},
$$

$$
T=\frac{2\pi}{12.65}
=0.4967\ \mathrm{s}.
$$

Suppose the measurement begins with $x_0=0.0800\ \mathrm{m}$ and
$v_0=-0.300\ \mathrm{m\,s^{-1}}$. The amplitude is

$$
A=
\sqrt{
(0.0800\ \mathrm{m})^2+
\left(
\frac{-0.300\ \mathrm{m\,s^{-1}}}
{12.65\ \mathrm{rad\,s^{-1}}}
\right)^2
}
=0.0834\ \mathrm{m}.
$$

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:

$$
\phi=\atanTwo(0.0237\ \mathrm{m},0.0800\ \mathrm{m})
=0.288\ \mathrm{rad}.
$$

The resulting equation is

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

Substitution at $t=0$ returns $x_0=0.0800\ \mathrm{m}$. Differentiation returns
$v_0=-0.300\ \mathrm{m\,s^{-1}}$. These two direct substitutions are more
reliable than an informal visual phase check.

The total energy can be calculated two ways:

$$
E
=\frac12kA^2
=\frac12(40.0)(0.0834)^2
=0.139\ \mathrm{J},
$$

$$
E
=\frac12kx_0^2+\frac12mv_0^2
=0.128\ \mathrm{J}+0.0113\ \mathrm{J}
=0.139\ \mathrm{J}.
$$

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

$$
v_{\max}=\omega_0A
=1.06\ \mathrm{m\,s^{-1}},
\qquad
a_{\max}=\omega_0^2A
=13.3\ \mathrm{m\,s^{-2}}.
$$

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
$\omega_0A$ violates the energy relation and requires a review of units, phase,
or stiffness.

Series and parallel systems offer a second worked comparison. Take
$k_1=30.0\ \mathrm{N\,m^{-1}}$, $k_2=60.0\ \mathrm{N\,m^{-1}}$, and
$m=0.500\ \mathrm{kg}$. For the series connection,

$$
k_{\rm series}
=\left(\frac{1}{30.0}+\frac{1}{60.0}\right)^{-1}
=20.0\ \mathrm{N\,m^{-1}},
$$

$$
T_{\rm series}
=2\pi\sqrt{\frac{0.500}{20.0}}
=0.993\ \mathrm{s}.
$$

For the parallel connection,

$$
k_{\parallel}=30.0+60.0
=90.0\ \mathrm{N\,m^{-1}},
$$

$$
T_{\parallel}
=2\pi\sqrt{\frac{0.500}{90.0}}
=0.468\ \mathrm{s}.
$$

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 $F=3.00\ \mathrm{N}$, the series system extends

$$
x_{\rm series}=\frac{F}{k_{\rm series}}
=0.150\ \mathrm{m}.
$$

Its individual extensions are $x_1=F/k_1=0.100\ \mathrm{m}$ and
$x_2=F/k_2=0.0500\ \mathrm{m}$, which add to the total. The parallel system
extends only $0.0333\ \mathrm{m}$. The force splits as
$F_1=k_1x=1.00\ \mathrm{N}$ and $F_2=k_2x=2.00\ \mathrm{N}$, 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:

$$
\left[\sqrt{\frac{m}{k}}\right]
=
\sqrt{
\frac{\mathrm{kg}}
{\mathrm{N\,m^{-1}}}
}
=
\sqrt{\mathrm{s^2}}
=\mathrm{s}.
$$

For torsional motion,

$$
\left[\sqrt{\frac{I}{\kappa}}\right]
=
\sqrt{
\frac{\mathrm{kg\,m^2}}
{\mathrm{N\,m\,rad^{-1}}}
}
=\mathrm{s},
$$

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
$0.500\ \mathrm{kg}$ 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 $q_0$:

$$
U(q)
=
U(q_0)
+\frac12k(q-q_0)^2
+\frac16c_3(q-q_0)^3
+\frac1{24}c_4(q-q_0)^4+\cdots.
$$

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

$$
F(q)
=
-k(q-q_0)
-\frac12c_3(q-q_0)^2
-\frac16c_4(q-q_0)^3-\cdots.
$$

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.

$$
% caption: Linear range and nonlinear restoring response. The straight line is the
% tangent whose slope $-k$ is the stiffness used in the harmonic equation; the curved
% force law departs from it beyond the marked interval, shifting the measured period
% and producing asymmetric turning points at larger displacement.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0.5,1.5)--(5.8,1.5) node[right] {displacement};
\draw[->,black] (3.1,0.4)--(3.1,2.8) node[above] {restoring force};
\draw[acc,very thick] (1.5,2.044)--(4.7,0.956);
\draw[black,very thick] plot[domain=1.0:5.2,samples=160] (\x,{1.5-0.34*(\x-3.1)-0.05*(\x-3.1)^3});
\draw[black,dashed] (2.2,0.6)--(2.2,2.4);
\draw[black,dashed] (4.0,0.6)--(4.0,2.4);
\node[acc] at (2.6,1.42) {linear};
\node[black] at (5.35,0.55) {curved};
\node[below] at (3.1,0.55) {center};
\end{tikzpicture}
$$

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 $15\ \mathrm{mm}$, 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

$$
m_{\rm eff}\approx m+\frac13m_s.
$$

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.

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

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

$$
E_i
=
\frac12mv_i^2+\frac12kx_i^2
$$

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

$$
x(t)=A\cos(\omega_0t+\phi),
$$

the argument of cosine must be dimensionless. $\omega_0t$ is dimensionless,
and $\phi$ is an angle expressed in radians. A frequency in hertz must be
converted with $\omega_0=2\pi f$. Substituting $f$ directly into a cosine that
expects angular frequency changes the period by $2\pi$. 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
$\omega_0$.

## Reporting and uncertainty

A numerical report should specify the model first:

$$
m\ddot x+kx=0,
\qquad
x(t)=A\cos(\omega_0t+\phi).
$$

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 $0.250\ \mathrm{kg}$ moving mass and
$40.0\ \mathrm{N\,m^{-1}}$ stiffness gave
$T=(0.497\pm0.003)\ \mathrm{s}$ over amplitudes from $10$ to
$15\ \mathrm{mm}$. 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

$$
T^2=\frac{4\pi^2m}{k}.
$$

Plotting $T^2$ against $m$ for several added masses gives a straight-line test
of the model. The slope is $4\pi^2/k$. 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

$$
k_{\rm dyn}=\frac{4\pi^2m}{T^2},
$$

a first-order relative uncertainty estimate is

$$
\frac{\sigma_k}{k}
\approx
\frac{\sigma_m}{m}
+2\frac{\sigma_T}{T}.
$$

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:

$$
E=\frac12kA^2,
\qquad
v_{\max}=\omega_0A.
$$

Phase controls the predicted state at a specified time but does not change total
energy. A trigger-time uncertainty $\sigma_t$ becomes phase uncertainty

$$
\sigma_\phi\approx\omega_0\sigma_t.
$$

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:

$$
\omega_0=(12.65\pm0.08)\ \mathrm{rad\,s^{-1}},
\qquad
T=(0.4967\pm0.0031)\ \mathrm{s},
$$

with the coordinate origin, amplitude interval, and uncertainty method in the
surrounding text. The two values should satisfy $\omega_0T=2\pi$ 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.