---
title: Pendulum Motion
module: Oscillations and Waves
moduleNumber: 7
lessonNumber: 13
order: 713
summary: >
  A pendulum keeps time only because, for small swings, gravity supplies a restoring
  torque proportional to the angle — and $T=2\pi\sqrt{L/g}$ then follows without the
  mass appearing at all. We derive that result, mark exactly which assumptions carry
  it (small angle, negligible pivot loss, a rigid support), then relax them: finite
  amplitude lengthens the period through an elliptic integral, and an extended body
  replaces $L$ with the ratio of its moment of inertia to its center-of-mass
  distance. How the period drifts with amplitude or pivot position is what diagnoses
  the geometric, damping, and distributed-mass corrections.
topics: [Oscillations]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 14 — Oscillations; §§14-1–14-3"
---


## Geometry and small-angle motion

A simple pendulum consists of a compact bob of mass $m$ suspended from a light,
inextensible support of length $L$. The idealization assumes a stationary frictionless
pivot, a uniform gravitational acceleration, planar motion, and a bob small enough
that its center of mass lies a distance $L$ from the pivot. Its state is specified by
the signed angular displacement $\theta$ from the downward vertical. Positive and
negative values distinguish the two sides of equilibrium. Arc displacement is
$s=L\theta$, provided $\theta$ is measured in radians.

The length in the model runs from the pivot axis to the bob's center of mass. A ruler
placed only along the string omits the bob radius; a ruler placed from a support hook
can include an unknown offset above the pivot axis. Those details matter in a period
measurement because $T$ scales as $\sqrt{L}$. A massive string, an extended bob, or a
support that bends under load changes the system from a simple pendulum into a
physical pendulum. The simple model remains valid when those changes are small
relative to the required accuracy.

$$
% caption: The ideal simple-pendulum geometry uses the distance from pivot axis to bob center as the length $L$. The signed angle is measured from the downward vertical, and the curved coordinate $s=L\theta$ follows the bob's arc rather than a horizontal projection.
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The tangential direction is the efficient direction for the equation of motion. The
support force has no tangential component because the support is radial. Gravity
contributes $-mg\sin\theta$ along the positive-$\theta$ tangent. Newton's second law
therefore gives the exact nonlinear equation

$$
mL\ddot\theta=-mg\sin\theta,
\qquad
\ddot\theta+\frac{g}{L}\sin\theta=0.
$$

Gravity and tangential inertia share the factor $m$, so mass cancels from the ideal
equation. The cancellation assumes a compact bob and light support. Redistributed
mass changes the moment of inertia and requires the physical-pendulum equation.

$$
% caption: Tangential and radial directions at a displaced bob. Gravity has the component $mg\sin\theta$ toward equilibrium along the arc; tension lies along the support and therefore does no tangential work.
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$$

The radial force balance separately checks the geometry. Taking inward along
the support as positive gives

$$
\mathcal T-mg\cos\theta=mL\dot\theta^2,
\qquad
\mathcal T=mL\dot\theta^2+mg\cos\theta,
$$

where $\mathcal T$ denotes the support tension. Speed raises tension through the
centripetal term. At a turning point $\dot\theta=0$, the tension is $mg\cos\theta_0$.
The support must remain taut, so an ordinary string cannot sustain a trajectory for
which the required tension becomes negative. Small oscillations satisfy that
condition easily; high-energy loop motion belongs to a different constrained-motion
problem.

Gravitational potential energy is most convenient with zero at the lowest point. The
bob rises by $L(1-\cos\theta)$, so

$$
U(\theta)=mgL(1-\cos\theta).
$$

A release from rest at angular amplitude $\theta_0$ has total mechanical energy
$E=mgL(1-\cos\theta_0)$. Conservation of energy gives a speed formula valid at
any amplitude:

$$
\frac12mL^2\dot\theta^2+mgL(1-\cos\theta)
=mgL(1-\cos\theta_0),
\qquad
\dot\theta^2=\frac{2g}{L}(\cos\theta-\cos\theta_0).
$$

Taking the square root permits either sign of angular velocity. The
positive sign describes motion toward increasing angle, while the negative sign
describes the return branch. Energy identifies the two turning points but does not
identify which branch the bob occupies; the sign of $\dot\theta$ completes the state.

$$
% caption: Exact gravitational potential energy for a pendulum, referenced to the lowest point. A horizontal energy level set by release angle intersects the curve at the two turning angles; the vertical gap between energy and potential is kinetic energy.
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The potential curve explains the restoring direction without a component calculation.
For positive $\theta$, $\d U/\d\theta=mgL\sin\theta$ is positive, and the generalized
force $-\d U/\d\theta$ points toward smaller angle. Its curvature at equilibrium,
$\d^2U/\d\theta^2=mgL$, is the linear restoring coefficient used in the
small-angle model. The exact curve is shallower than a parabola at larger angles,
which lengthens the period.


**The small-angle oscillator.**

For angles expressed in radians,

$$
\sin\theta=\theta-\frac{\theta^3}{6}+\frac{\theta^5}{120}-\cdots.
$$

Replacing $\sin\theta$ by $\theta$ retains the first term. The resulting equation
has constant coefficients,

$$
\ddot\theta+\omega_0^2\theta=0,
\qquad
\omega_0=\sqrt{\frac{g}{L}},
\qquad
T_0=2\pi\sqrt{\frac{L}{g}}.
$$

The subscript on $T_0$ marks the zero-amplitude limit. Finite swings have an
amplitude-dependent period because the exact restoring torque has smaller magnitude
than the linear approximation at the same nonzero angle. Report the amplitude range
whenever the required accuracy is better than a few parts in $10^3$.

$$
% caption: The curve $\sin\theta$ lies below the straight small-angle approximation for positive angle and above it for negative angle. Their agreement near the origin explains why the linear model describes sufficiently small swings, while the growing gap predicts a longer finite-amplitude period.
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$$

The force-level relative error of the replacement is

$$
\varepsilon_{\sin}
=\frac{\theta-\sin\theta}{\theta}
\approx\frac{\theta^2}{6}
\quad (\theta\ne0).
$$

At $10^\circ$ ($0.1745\ \mathrm{rad}$), that local force error is about $0.51\%$.
At $20^\circ$, it is about $2.0\%$; at $30^\circ$, about $4.5\%$. The period error is
substantially smaller than the instantaneous force error because a period averages
motion through the whole arc. The two quantities answer different questions and
should not be interchanged in an uncertainty statement.

The linear solution can be written in several equivalent forms. With angular
amplitude $\Theta$ and phase constant $\delta$,

$$
\theta(t)=\Theta\cos(\omega_0t+\delta),
\qquad
\dot\theta(t)=-\omega_0\Theta\sin(\omega_0t+\delta),
\qquad
\ddot\theta(t)=-\omega_0^2\theta(t).
$$

Angular speed reaches its maximum magnitude at equilibrium and vanishes at the
turning points. Angular acceleration has the opposite sign to displacement and its
largest magnitude at a turning point. A fitted acceleration maximum at equilibrium
indicates an error in the coordinate or phase convention.

Initial angle $\theta_i$ and initial angular velocity $u_i$ determine amplitude and
phase without ambiguity. Expressing the solution as

$$
\theta(t)=C\cos\omega_0t+D\sin\omega_0t
$$

gives $C=\theta_i$ and $D=u_i/\omega_0$. Hence

$$
\Theta=\sqrt{\theta_i^2+\left(\frac{u_i}{\omega_0}\right)^2},
\qquad
\tan\delta=-\frac{u_i}{\omega_0\theta_i},
$$

with the quadrant selected from the signs of $\theta_i$ and $u_i$. The two-argument
angle function $\atanTwo(-u_i,\omega_0\theta_i)$ records that quadrant safely when
software is used. Dividing by $\theta_i$ before retaining its sign produces a phase
shift of half a cycle in two quadrants.

The small-angle period has a dimensional form. Only $L$ and $g$ appear, and
$L/g$ has units of time squared. Doubling length multiplies period by $\sqrt2$;
quadrupling length doubles period. The bob mass does not enter. A data set in which
period changes appreciably after swapping equally shaped bobs points to a changed
length, a changing pivot, air drag acting on a large bob, or a measurement artifact
rather than to the ideal formula.


## Energy, phase, and finite-amplitude effects

The exact energy balance separates pendulum geometry from the
small-angle approximation. Kinetic energy is rotational kinetic energy about the
pivot, $K=\tfrac12mL^2\dot\theta^2$, and the potential energy is the cosine function
above. Therefore

$$
E=\frac12mL^2\dot\theta^2+mgL(1-\cos\theta).
$$

At small angle, $1-\cos\theta\approx\theta^2/2$. The energy reduces to the quadratic
form

$$
E\approx\frac12mL^2\dot\theta^2+\frac12mgL\theta^2
=\frac12mL^2\dot\theta^2+\frac12mL^2\omega_0^2\theta^2.
$$

The first term is largest at equilibrium and the second is largest at the turning
angles. Each quadratic contribution has time average $E/2$ in the linear model.
Instantaneous kinetic and potential energies exchange continuously and remain a
quarter cycle out of phase.

$$
% caption: Linearized kinetic and potential energies over one cycle. Potential energy is largest at turning angles, kinetic energy is largest at equilibrium crossings, and their sum remains at the horizontal total-energy level.
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$$

Substitution of the linear solution yields the phase-dependent forms below.

$$
U\approx\frac12mgL\Theta^2\cos^2(\omega_0t+\delta),
\qquad
K\approx\frac12mgL\Theta^2\sin^2(\omega_0t+\delta),
$$

and

$$
E\approx\frac12mgL\Theta^2.
$$

The amplitude appears squared. Doubling a release angle in the linear range
quadruples stored energy and doubles maximum speed. A video data set with a
half-amplitude trace should therefore have one quarter of the energy scale if its
length and bob mass are unchanged. Period, by contrast, remains unchanged only in
the linear approximation.

At an arbitrary angle, conservation of energy gives the exact angular speed quoted
earlier. Its maximum is at $\theta=0$:

$$
\dot\theta_{\max}
=\sqrt{\frac{2g}{L}(1-\cos\theta_0)},
\qquad
v_{\max}
=\sqrt{2gL(1-\cos\theta_0)}.
$$

For small amplitude, these become $\dot\theta_{\max}\approx\omega_0\Theta$ and
$v_{\max}\approx\sqrt{gL}\,\Theta$. The speed formula uses a center-of-mass arc
speed. A horizontal image coordinate from a camera differs from arc displacement by
geometric projection, especially away from the vertical. Calibrate an image-based
analysis with angular position or with $x=L\sin\theta$, then convert deliberately.

The tension measurement can also test the energy calculation. Combining
$\dot\theta^2=2g(\cos\theta-\cos\theta_0)/L$ with the radial equation yields

$$
\mathcal T
=mg\bigl(3\cos\theta-2\cos\theta_0\bigr).
$$

At the bottom, $\mathcal T_{\rm bottom}=mg(3-2\cos\theta_0)$. At small angle, the
tension increment over weight is $mg\theta_0^2$, to leading order. A load cell
at the pivot must resolve a difference much smaller than the mean static load in a
small-amplitude experiment, which often makes video tracking more practical for
student-scale apparatus.

The linear phase portrait has an elliptical energy relation.

$$
\left(\frac{\theta}{\Theta}\right)^2
+\left(\frac{\dot\theta}{\omega_0\Theta}\right)^2=1.
$$

The top and bottom of the ellipse represent equilibrium crossings with opposite
velocity. The left and right ends represent turning positions. The direction of
motion around the curve follows the sign convention; mark a measured
sequence of states rather than treating the ellipse as an unlabeled shape. Damping
would contract the ellipse over time, while external driving would require a separate
forced-oscillation treatment.

$$
% caption: A small-amplitude pendulum follows a closed energy ellipse in angular displacement and angular speed. Arrows identify the temporal direction: release from the right turning point passes through positive speed at equilibrium before reaching the left turning point.
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> **Worked example.** A bob of mass $m=0.250\ \mathrm{kg}$ hangs from a pendulum of effective length
> $L=0.800\ \mathrm{m}$, released from rest at $\theta_0=0.120\ \mathrm{rad}$. The
> small-angle period is
>
> $$
> T_0=2\pi\sqrt{\frac{0.800\ \mathrm{m}}{9.81\ \mathrm{m\,s^{-2}}}}
> =1.794\ \mathrm{s}.
> $$
>
> At the bottom of the trajectory, maximum speed is
>
> $$
> v_{\max}
> =\sqrt{2(9.81\ \mathrm{m\,s^{-2}})(0.800\ \mathrm{m})
> \bigl[1-\cos(0.120)\bigr]}
> =0.336\ \mathrm{m\,s^{-1}}.
> $$
>
> The bottom tension is
>
> $$
> \mathcal T_{\rm bottom}
> =0.250\ \mathrm{kg}\,[9.81+0.336^2/0.800]\ \mathrm{m\,s^{-2}}
> =2.49\ \mathrm{N}.
> $$
>
> The finite-amplitude period shift at $0.120\ \mathrm{rad}$ is about $\theta_0^2/16
> =9.0\times10^{-4}$, or $0.090\%$. It is negligible for a stopwatch result reported
> to $0.01\ \mathrm{s}$, but it exceeds a target uncertainty of $10^{-4}$ in a timed
> sensor experiment. Experimental adequacy follows from the stated tolerance, not
> from an unqualified label such as “small angle.”


**Finite amplitude and the nonlinear period.**

The exact equation does not have a cosine solution at finite amplitude. Energy gives
the time increment instead:

$$
\d t=\sqrt{\frac{L}{2g}}\,
\frac{\d\theta}{\sqrt{\cos\theta-\cos\theta_0}}.
$$

One quarter of a cycle carries the bob from $\theta_0$ to zero, so the exact period is

$$
T=4\sqrt{\frac{L}{2g}}
\int_0^{\theta_0}
\frac{\d\theta}{\sqrt{\cos\theta-\cos\theta_0}}.
$$

The integrand grows near the release angle because velocity approaches zero there.
The divergence remains integrable because the bob spends finite time near the
turning point. The integral contains the full nonlinear dynamics. Replacing the
cosine before integration removes the amplitude dependence and recovers $T_0$.

With $k=\sin(\theta_0/2)$ and the substitution
$\sin(\theta/2)=k\sin\varphi$, the exact expression becomes

$$
T=4\sqrt{\frac{L}{g}}\,K(k),
\qquad
K(k)=\int_0^{\pi/2}\frac{\d\varphi}{\sqrt{1-k^2\sin^2\varphi}},
$$

where $K$ is the complete elliptic integral of the first kind. Its small-$k$ series
gives a compact correction formula:

$$
\frac{T}{T_0}
=1+\frac14k^2+\frac{9}{64}k^4+\frac{25}{256}k^6+\cdots
=1+\frac{\theta_0^2}{16}+\frac{11\theta_0^4}{3072}+\cdots.
$$

The second equality is a series in radians. The leading correction is positive, so a
finite-amplitude pendulum runs slower than its infinitesimal-amplitude calibration.
At $10^\circ$, the leading fractional shift is approximately $1.90\times10^{-3}$;
at $30^\circ$, it is approximately $1.71\times10^{-2}$. Higher terms are needed as
the required uncertainty approaches those omitted contributions.

$$
% caption: Exact period ratio versus release amplitude rises above the small-angle value of one. The dashed tangent-scale reference highlights the initially quadratic rise; visible curvature at larger amplitude comes from higher terms in the elliptic-integral expansion.
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The leading term sets an amplitude guideline for a period experiment with allowable
fractional model bias $b$,

$$
\frac{\theta_0^2}{16}\lesssim b,
\qquad
\theta_0\lesssim4\sqrt b.
$$

For $b=10^{-3}$, the guideline is $\theta_0\lesssim0.126\ \mathrm{rad}$, about
$7.2^\circ$. For $b=10^{-2}$, it is $\theta_0\lesssim0.400\ \mathrm{rad}$, about
$23^\circ$. The inequality reserves the full error budget for amplitude alone. A
measurement with sizable length, timing, and temperature contributions should adopt
a tighter amplitude cap.

Amplitude also changes during an ordinary free swing because of air resistance and
pivot losses. Even weak damping produces a slow amplitude decrease; the nonlinear
period therefore drifts downward toward $T_0$. A period average over many cycles
then mixes several amplitudes. Record the initial and final angle, use a short timing
window, or fit a model that includes amplitude evolution. The separate damped-
oscillator lesson treats the decay law and quality factor; its formulas should not be
inserted into an undamped period fit without checking the apparatus.

At an angular displacement $\theta$, the bob moves on a circle of radius $L$. A
tangential displacement and velocity have the signs of $\theta$ and $\dot\theta$,
respectively:

$$
s=L\theta,
\qquad
v_t=L\dot\theta,
\qquad
a_t=L\ddot\theta.
$$


## Physical pendulums and mass distribution

A physical, or compound, pendulum is a rigid body free to rotate about a horizontal
axis that does not pass through its center of mass. The body may be a rod, a plate,
a meter rule with added masses, or a shaped laboratory object. Let $M$ be its total
mass, $D$ the distance from pivot axis to center of mass, and $I_P$ the moment of
inertia about the pivot. A displacement $\theta$ produces gravitational torque

$$
\tau_P=-MgD\sin\theta.
$$

Rotational Newton's second law gives the exact equation

$$
I_P\ddot\theta+MgD\sin\theta=0.
$$

The motion has the simple-pendulum form, with the full moment of inertia replacing
the point-bob term. Mass far from the pivot contributes strongly to $I_P$ even when
its contribution to center-of-mass displacement is small. The period therefore
depends on the distribution of mass as well as its total amount.

$$
% caption: A physical pendulum rotates about an axis separated by distance $D$ from its center of mass. The weight acts at the center of mass, so its lever arm is $D\sin\theta$ and the restoring torque is proportional to that lever arm.
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For small angular displacement, replacing $\sin\theta$ by $\theta$ gives

$$
I_P\ddot\theta+MgD\theta=0,
\qquad
\omega_0=\sqrt{\frac{MgD}{I_P}},
\qquad
T_0=2\pi\sqrt{\frac{I_P}{MgD}}.
$$

Total mass cancels when $I_P$ is written as $M$ times a geometric quantity. The
cancellation has the same origin as in a simple pendulum, but the geometry contains
more than one length. Two rigid bodies with the same $M$, $D$, and external outline
can have different periods if internal mass is rearranged. Conversely, scaling every
mass in a rigid body by the same factor leaves its small-angle period unchanged.

The parallel-axis theorem converts a tabulated center-of-mass moment of inertia into
the pivot moment:

$$
I_P=I_{\rm cm}+MD^2.
$$

Define the radius of gyration about a parallel center-of-mass axis by
$I_{\rm cm}=Mk_G^2$. The period then has a purely geometric form,

$$
T_0=2\pi\sqrt{\frac{k_G^2+D^2}{gD}}.
$$

The numerator contains two mechanisms. The term $D^2$ is the inertia that the whole
mass would have if concentrated at the center of mass. The term $k_G^2$ is extra
inertia from spread around that center. Treating an extended object as a point mass
at its center of mass drops the latter term and generally predicts a period that is
too short.

An equivalent simple-pendulum length packages the physical pendulum into one
effective length,

$$
L_{\rm eq}=\frac{I_P}{MD}=D+\frac{k_G^2}{D},
\qquad
T_0=2\pi\sqrt{\frac{L_{\rm eq}}{g}}.
$$

Equivalent length preserves the small-angle period of a corresponding simple
pendulum. A long distributed object can have an equivalent length beyond its far
end because the value equals the ratio of inertia to gravitational torque, rather
than the location of a marked point on the body.

### Uniform rod about a transverse pivot

A uniform thin rod of length $\ell$ pivoted at one end has center of mass at
$D=\ell/2$ below the axis and pivot moment of inertia
$I_P=\tfrac13M\ell^2$. Substitution gives

$$
T_0
=2\pi\sqrt{\frac{\tfrac13M\ell^2}{Mg(\ell/2)}}
=2\pi\sqrt{\frac{2\ell}{3g}},
\qquad
L_{\rm eq}=\frac{2\ell}{3}.
$$

The equivalent length is longer than the center-of-mass distance $\ell/2$ because
parts of the rod lie farther from the axis. A $0.900\ \mathrm{m}$ uniform rod has
predicted period $1.55\ \mathrm{s}$. A measured value far from that prediction
may indicate a pivot offset, nonuniform mass, finite-amplitude bias, or an end support
that adds a substantial concentrated mass.

$$
% caption: A uniform rod pivoted at one end has center-of-mass distance one half of its length and pivot moment $M\ell^2/3$. Its equivalent simple-pendulum length is two thirds of the rod length, between the center of mass and the far end.
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A uniform rod pivoted a distance $x$ from its center has
$D=|x|$ and $I_P=\tfrac1{12}M\ell^2+Mx^2$. Its period is

$$
T_0(x)=2\pi\sqrt{\frac{\ell^2/12+x^2}{g|x|}}.
$$

As $x$ tends to zero, gravity has almost no lever arm and the period becomes very
large. As $|x|$ becomes large, the physical pendulum approaches a point-like bob at
distance $|x|$, so the period also grows. Differentiating the quantity under the
square root gives a minimum at

$$
|x|=\frac{\ell}{\sqrt{12}}\approx0.289\ell.
$$

The minimum occurs where the reduction in rotational inertia from moving the pivot
toward the center balances the decrease in the restoring-torque coefficient
$Mg|x|$. It guides the design of a period-sensitive rod pendulum, while gravity
measurements may prioritize a readily measured effective length.

$$
% caption: Period of a uniform rod versus pivot distance from its center of mass. The curve diverges at a center pivot because gravitational torque vanishes there, has a finite minimum, and rises again when the pivot moves far from the rod.
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$$

**Inferring a moment of inertia from a period.**

If $M$, $D$, and the small-angle period are known, period data determine a pivot
moment of inertia:

$$
I_P=MgD\left(\frac{T_0}{2\pi}\right)^2.
$$

This calculation handles an object whose shape makes a direct integral awkward.
The result is sensitive to the square of period, so a fractional period uncertainty
contributes twice as strongly to fractional $I_P$ uncertainty. Center-of-mass
location enters linearly through $D$ and must be measured from the actual pivot axis.
A knife edge with a broad contact region requires a clear operational definition of
the axis; otherwise the inferred inertia has an uncontrolled systematic offset.


## Gravity measurement and apparatus control

The local gravitational acceleration is

$$
g=\frac{4\pi^2L}{T_0^2}.
$$

Experimental accuracy depends on the definitions of $L$ and $T$. Length is the
pivot-axis-to-center-of-mass distance. A spherical bob has its
center at its geometric center only when it is homogeneous. A cylindrical bob hung
from one end can have a center several centimeters from its visible midpoint. A
support cord that stretches changes the effective length between the static setup and
the oscillating state. Record the measuring endpoints and apparatus configuration in
the data sheet rather than leaving the length definition implicit.

The period is the elapsed time for one complete return to the same state. A
reproducible operational state is an equilibrium crossing in a specified direction. Timing from a
leftward crossing to the next leftward crossing avoids ambiguity about half cycles.
The first release is a poor trigger when fingers perturb the bob or when the release
angle is estimated visually. A light gate, video timestamp, or magnetic sensor near
the bottom offers a repeatable event. Keep the sensor's threshold location unchanged
throughout a series.

Timing $N$ cycles reduces the effect of a start-stop timing uncertainty on the
estimated period. If the total elapsed time is $t_N$,

$$
T=\frac{t_N}{N},
\qquad
g=\frac{4\pi^2LN^2}{t_N^2}.
$$

A fixed absolute uncertainty $u(t_N)$ contributes $u(T)=u(t_N)/N$ to a single
period. More cycles reduce the random or resolution-limited
endpoint contribution. Cycle-count errors, amplitude drift, clock calibration, and
unequal trigger delays require separate control. Select $N$ by balancing timing
resolution against drift and missed-cycle risk.

For independent small standard uncertainties, first-order propagation gives

$$
\left(\frac{u(g)}{g}\right)^2
=\left(\frac{u(L)}{L}\right)^2
+\left(2\frac{u(T)}{T}\right)^2.
$$

The factor of two on period follows from the inverse-square relation. A fractional
period error of $5\times10^{-4}$ contributes $10^{-3}$ to the fractional uncertainty
in $g$. The expression concerns standard uncertainties from specified random or
calibration components. A suspected one-sided bias from finite amplitude is not made
safe by inserting it as though it were random scatter; correct the model or state a
bounded systematic contribution separately.

> **Worked example.** Suppose the effective length is measured as
> $L=(0.9974\pm0.0005)\ \mathrm{m}$. A sensor records $N=50$ complete cycles in
> $t_N=(100.27\pm0.10)\ \mathrm{s}$. The mean period and gravity estimate are
>
> $$
> T=\frac{100.27\ \mathrm{s}}{50}=2.0054\ \mathrm{s},
> $$
>
> $$
> g=\frac{4\pi^2(0.9974\ \mathrm{m})}{(2.0054\ \mathrm{s})^2}
> =9.791\ \mathrm{m\,s^{-2}}.
> $$
>
> Treating the stated endpoint and length uncertainties as independent standard
> uncertainties gives
>
> $$
> \frac{u(g)}{g}
> =\sqrt{\left(\frac{0.0005}{0.9974}\right)^2
> +\left(2\frac{0.10}{100.27}\right)^2}
> =2.06\times10^{-3},
> $$
>
> so $u(g)=0.020\ \mathrm{m\,s^{-2}}$ before model-bias allowances. The numerical
> result should be reported as $g=(9.79\pm0.02)\ \mathrm{m\,s^{-2}}$ only when the
> quoted uncertainty convention has been stated and the release amplitude, length
> definition, and clock calibration have passed their own checks. Rounding a result to
> many digits does not compensate for unexamined systematic effects.

### Multiple lengths and the $T^2$ relation

A series of pendulum lengths separates a fixed length offset from a single-measurement
error.
For small amplitudes,

$$
T^2=\frac{4\pi^2}{g}L_{\rm eff}.
$$

If a measured string length $L_m$ omits a constant pivot-to-reference or
reference-to-center offset $b$, then $L_{\rm eff}=L_m+b$ and

$$
T^2=aL_m+c,
\qquad
a=\frac{4\pi^2}{g},
\qquad
c=\frac{4\pi^2b}{g}.
$$

The slope determines $g=4\pi^2/a$; the intercept estimates the constant length
offset. Curvature in the $T^2$ plot calls for diagnosis before reporting a
slope. Potential causes include changing release amplitude, a cord that stretches
by different amounts, support compliance, or a calibration offset that is not
constant across lengths.

$$
% caption: Squared period is linear in effective length for small oscillations. A nonzero intercept on a ruler-based length plot can indicate a constant unmeasured offset between the ruler reference and the true center-of-mass length.
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$$

Use repeated timing trials at each length. Let $T_{ij}$ denote the period found from
trial $j$ at length $i$. A preliminary mean and repeatability measure are

$$
\bar T_i=\frac1r\sum_{j=1}^{r}T_{ij},
\qquad
s_{\bar T_i}=\frac{1}{\sqrt r}
\sqrt{\frac{\sum_j(T_{ij}-\bar T_i)^2}{r-1}}.
$$

When timing precision differs strongly among lengths, fit $T_i^2$ with weights based
on its uncertainty rather than treating every point as equally precise. A common
length calibration error is shared by all points and is correlated; it cannot be
reduced by adding more timing trials. Carry shared calibration terms separately when
computing the slope uncertainty.

A residual plot evaluates the straight-line relation. Define

$$
r_i=T_i^2-(aL_{m,i}+c).
$$

Random residuals centered around zero are consistent with the fitted model, subject
to the size expected from stated uncertainty. A curved residual pattern is evidence
against a single constant slope and intercept. A monotonic pattern ordered by trial
time can indicate amplitude decay or temperature drift. Residual structure indicates
model mismatch; the apparatus and acquisition record constrain its likely cause.


**Model bias, apparatus checks, and controlled acquisition.**

Validate the pendulum model before combining an uncertainty budget. Separate random
scatter from directional shifts. The principal model and apparatus contributions are:

- **Finite amplitude.** The exact period exceeds $T_0$ by approximately
  $\theta_0^2/16$. Measure the release angle in radians, reduce it, or apply a
  stated correction with an uncertainty for the angle.
- **Length definition.** Pivot width, bob center location, suspension stretch, and
  a ruler zero offset change $L$. These terms generally shift every trial in the
  same direction.
- **Timing event.** Human reaction, camera frame quantization, sensor threshold,
  and a missed cycle affect $t_N$. Repeating one mechanical event with the same
  direction limits ambiguity.
- **Dissipation and support motion.** Drag changes amplitude during a long run;
  friction and a moving support can alter both period and the assumed equilibrium.
- **Three-dimensional motion.** An elliptical or conical bob path changes the
  projected angle and can couple horizontal directions.

Estimate each term on the scale of the desired fractional uncertainty. A $0.5\ \mathrm{mm}$
length ambiguity on a one-meter pendulum is already $5\times10^{-4}$ in $L$, while
a low-resolution stopwatch may dominate a short timing run by much more.

**Amplitude control and correction.**

Release angle is measured from the vertical, not from the horizontal or from a
screen-edge coordinate. A plumb line through the pivot marks the vertical.
The geometry of lateral displacement $x$ measured at the bob center gives

$$
\theta_0=\arcsin\left(\frac{x}{L}\right),
$$

provided $x$ is perpendicular to the vertical line in the swing plane. The simpler
ratio $x/L$ is a small-angle approximation. A hand-held protractor can be adequate
at several degrees; video calibration must account for camera perspective and the
plane of the swing. Mark an amplitude limit on the apparatus so every trial begins
within the model-bias allowance.

If a finite-amplitude correction is appropriate, use the same angle convention in
the correction and uncertainty propagation. To leading order,

$$
T\approx T_0\left(1+\frac{\theta_0^2}{16}\right),
\qquad
T_0\approx\frac{T}{1+\theta_0^2/16}.
$$

Small independent angle uncertainty $u(\theta_0)$ gives correction-factor
uncertainty approximately $\theta_0u(\theta_0)/8$. The approximation applies
only when higher amplitude terms are smaller than the required error. At larger
angles, evaluate the elliptic integral numerically or reduce the amplitude instead
of extending a truncated series beyond its validated range.

**Length stability and pivot geometry.**

Measure the length after the bob and support have reached their operating load. A
thin fiber under a static bob can lengthen with temperature or humidity; a metal rod
expands with temperature; a flexible clamp can rotate under changing tension. For a
small uniform thermal expansion coefficient $\alpha$,

$$
\frac{\Delta L}{L}\approx\alpha\Delta\mathcal T_{\rm env},
\qquad
\frac{\Delta T}{T}\approx\frac12\alpha\Delta\mathcal T_{\rm env},
$$

where $\Delta\mathcal T_{\rm env}$ is a temperature change, not the support tension
$\mathcal T$. The notation distinction matters because both quantities occur in
pendulum work. A temperature correction is rarely the dominant term in a short
classroom trial, but it can matter in a long comparison or in a high-resolution
clock. Record ambient conditions when a length calibration is reused.

An extended bob can be treated as a physical pendulum if its size is no longer
negligible. A dense spherical bob of radius $r$ attached to a light string of length
$\ell_s$ has center distance $D\approx\ell_s+r$, but its own moment of inertia makes
the exact compound-pendulum period differ slightly from the point-bob formula.
The correction is typically of order $(r/D)^2$ for a compact sphere. State whether
the quoted length and model neglect that term; a millimeter-scale bob on a meter
string and a large hollow bob on a short cord do not share the same approximation
quality.

**Planarity, support motion, and effective gravity.**

The derivation assumes the bob moves in one vertical plane. A sideways release gives
two horizontal components and can produce a slowly rotating elliptical path. The
observed projection may then have a period near the pendulum period while its angle
calibration is wrong. Start from rest without an azimuthal push, use a thin guide only
when it does not rub the support, and inspect video from a direction normal to the
intended plane.

A support accelerating horizontally with acceleration $\vec a_0$ changes the
equilibrium direction in the support frame. The effective gravitational acceleration
is

$$
\vec g_{\rm eff}=\vec g-\vec a_0,
\qquad
g_{\rm eff}=\sqrt{g^2+a_0^2}
\quad\text{for horizontal }\vec a_0,
$$

and the equilibrium tilt obeys $\tan\theta_{\rm eq}=a_0/g$. Small oscillations about
that tilted direction have period $2\pi\sqrt{L/g_{\rm eff}}$. Vibration near the
pendulum frequency can contaminate timing even when the support's mean acceleration
is zero. Keep the pivot on a rigid, isolated support and record disturbances during
acquisition.

$$
% caption: In a horizontally accelerating support frame, gravity and the inertial contribution combine into an effective downward direction. The pendulum aligns with that resultant and oscillates about the tilted equilibrium rather than about the geometric vertical.
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### A defensible acquisition sequence

Prepare the apparatus before collecting a production run.

1. Establish the pivot axis and bob center, then measure the effective length with
   the pendulum hanging at rest. Record ruler resolution, calibration, and the
   endpoint convention.
2. Mark a vertical reference and a maximum release angle derived from the allowed
   finite-amplitude bias. Release from rest without a lateral push.
3. Select one repeatable timing event and its direction. Test the sensor or video
   trigger for several crossings before beginning the timed run.
4. Count a predetermined number of complete cycles. Keep a separate manual count
   or saved event log so a missed cycle can be detected later.
5. Repeat the total-time measurement. Record initial and final amplitude, ambient
   changes, visible support motion, and any rejected run with its reason.
6. Reduce each run to period, inspect repeated values, propagate random components,
   then add model and calibration contributions using an explicitly stated method.

Keep the original event record and the criterion for every rejected run. An
unexplained deletion changes the reported sample and can conceal a timing or release
fault. A documented outlier investigation can justify exclusion.

> **Worked example.** Four ruler-based settings are $L_m=0.400$, $0.600$, $0.800$, and
> $1.000\ \mathrm{m}$. Suppose repeated low-amplitude measurements give mean
> values of $T^2$ near $1.660$, $2.465$, $3.271$, and $4.077\ \mathrm{s^2}$.
> Their straight-line fit has slope approximately
>
> $$
> a=4.03\ \mathrm{s^2\,m^{-1}},
> \qquad
> c=0.048\ \mathrm{s^2}.
> $$
>
> The inferred quantities are
>
> $$
> g=\frac{4\pi^2}{a}\approx9.80\ \mathrm{m\,s^{-2}},
> \qquad
> b=\frac{c}{a}\approx0.012\ \mathrm{m}.
> $$
>
> The $12\ \mathrm{mm}$ intercept-derived offset is plausible for a ruler reference
> that begins at the bottom of a support fitting rather than at the pivot axis. The
> single-length result could have absorbed that offset into a biased gravity estimate.
> The line fit identifies it because changing $L_m$ changes the slope contribution but
> leaves the constant geometry offset unchanged. A residual inspection remains
> essential; the numerical agreement of four points with a line is evidence about the
> model only to the resolution and controlled conditions of that data set.


## Sensitivity, diagnostics, and model scope

Logarithmic differentiation yields a compact sensitivity
relation:

$$
\frac{\d T_0}{T_0}
=\frac12\frac{\d L}{L}-\frac12\frac{\d g}{g}.
$$

A one-percent increase in effective length raises period by one half percent. A
one-percent increase in gravitational acceleration lowers period by one half percent.
The relation guides experimental design. A longer pendulum has a longer period, so a
given timestamp resolution is a smaller fraction of one cycle. Its greater physical
size can make length measurement, room clearance, support rigidity, and planar
release more difficult. An optimal design balances these actual limits rather than
maximizing $L$ automatically.

Known local gravity calibrates a length through

$$
L_{\rm eff}=\frac{gT_0^2}{4\pi^2}.
$$

The result is a dynamical length that reproduces the observed small-angle period.
Additional geometry is required to separate a bob-center offset, suspension stretch,
and physical-pendulum inertia correction. Likewise, a physical-pendulum period
determines a ratio of inertia to restoring torque until mass and center-of-mass
position are measured independently.

### Reading common data patterns

Measured period is expected to be symmetric with respect to the sign of release
angle in the ideal model: releases at $+\theta_0$ and $-\theta_0$ have the same
period. A repeatable difference between sides points to asymmetric pivot friction,
a support geometry issue, a lateral disturbance, or an angle reference that is not
vertical. Measure both directions when an apparatus allows it. The comparison tests
more than random scatter because it reverses a controlled sign while retaining the
same nominal energy.

Period should increase quadratically with the magnitude of a controlled release
angle near zero. A straight-line dependence on signed angle is therefore evidence of
an asymmetric mechanism or analysis error, not the expected nonlinear pendulum
correction. Plotting period against $\theta_0^2$ is often more diagnostic than
plotting it against angle itself. The intercept estimates the zero-amplitude period;
the slope can be compared with the $T_0/16$ leading prediction only after length and
angle calibration have been checked.

A length sweep has a different signature. The $T^2$ versus $L_m$ plot should be
linear. A constant offset changes the intercept but preserves slope. A systematic
curvature can arise when the support stretches more at long settings, when amplitude
is not held constant, or when a pendulum that appears simple has a changing physical
geometry. A few points near a straight line leave those alternatives unresolved.
Inspect repeated trials, residuals, and the independent length measurements.

> **Worked example.** A pendulum timing mechanism calibrated at a nonzero amplitude changes rate as its
> amplitude decays. At $\theta_0=10.0^\circ=0.1745\ \mathrm{rad}$, the leading period
> correction is
>
> $$
> \frac{T(10^\circ)-T_0}{T_0}
> \approx\frac{(0.1745)^2}{16}
> =1.90\times10^{-3}.
> $$
>
> If the mechanism is calibrated while the swing has that amplitude and subsequently
> operates close to zero amplitude, each cycle is shorter by about $0.190\%$. A clock
> that counts those cycles gains approximately
>
> $$
> (1.90\times10^{-3})(86400\ \mathrm{s})
> \approx164\ \mathrm{s}
> \approx2.7\ \mathrm{min}
> $$
>
> per day, before other clock errors. Precision pendulum systems constrain amplitude
> or apply a nonlinear correction. A period result must state its amplitude range.

**Compact reporting standard.**

A complete pendulum result identifies the model and the evidence used to validate it.
Include the effective-length definition, bob and support description, release-angle
range, number of timed cycles, event definition, timing instrument, repeated-trial
method, and uncertainty convention. A multi-length fit also requires plotted points,
fit equation, residuals, treatment of correlated length calibration, and any excluded
run with an objective criterion. A physical-pendulum report adds the pivot axis,
center-of-mass determination, and whether the inferred quantity is $I_P$ or an
equivalent length.

Separate the numerical estimate from its accuracy claim. A period measurement can be
repeatable while biased by a center-of-mass offset; it can agree with a reference
through chance cancellation between amplitude and length errors; it can have a
narrow fit confidence interval while a support mode violates the assumed
one-degree-of-freedom model. The reported value should trace back to apparatus
geometry, event timing, and stated model checks.


**Limit checks and comparison tests.**

Several limits catch algebraic and modeling errors before numerical data are used.
A point bob with $\theta_0\to0$ in the exact elliptic-integral expression must give
$K(0)=\pi/2$ and hence $T\to2\pi\sqrt{L/g}$. A physical pendulum concentrated at
distance $D$ from its pivot has $I_P\to MD^2$ and period $2\pi\sqrt{D/g}$. An
extended body with $D\to0$ has vanishing gravitational torque while $I_P$ remains
finite, so its small-angle period diverges.
Each limit has a clear physical interpretation and exposes a different incorrect
substitution.

At the same center-of-mass distance $D$, compare a physical pendulum with a point
bob placed at that distance. The period ratio is

$$
\frac{T_{\rm physical}}{T_{\rm point}(D)}
=\sqrt{\frac{I_P}{MD^2}}
=\sqrt{1+\frac{k_G^2}{D^2}}.
$$

The ratio exceeds one whenever the rigid body has nonzero mass spread. A measured
ratio below one signals an inconsistent distance, an incorrect moment-of-inertia
formula, or a period comparison made at different amplitudes. The form also gives a
dimensionless criterion for the point-bob approximation: $k_G/D\ll1$ keeps the
fractional period change near $k_G^2/(2D^2)$.

Unit checks complement the limit checks. The simple-pendulum relation has
$[L/g]=\mathrm{s^2}$, the physical-pendulum relation has
$[I_P/(MgD)]=\mathrm{s^2}$, and the gravity estimator has units
$\mathrm{m\,s^{-2}}$. Angles in the sine series and elliptic-integral correction
are dimensionless radians. Inserting degrees directly into $\theta_0^2/16$ inflates
the correction by roughly $(180/\pi)^2$. Convert angle measurements before fitting,
correction, or uncertainty propagation.

**Sampling and event interpolation.**

Digital acquisition has a sample interval
$\Delta t=1/f_s$, where $f_s$ is camera or sensor rate. If an equilibrium crossing
is assigned to the nearest video frame, an individual timestamp has a quantization
scale of roughly $\Delta t/\sqrt{12}$ under a uniform rounding model. Independent
start and stop assignments give a total-time scale near $\Delta t/\sqrt6$ before
including tracking noise. A many-cycle run has associated gravity contribution
approximately

$$
\frac{u(g)}{g}\approx2\frac{\Delta t/\sqrt6}{t_N}.
$$

The fractional timing bound compares sample rate and cycle count on the same scale.
A $30\ \mathrm{Hz}$ recording can support a coarse period measurement over
many cycles; it is poorly suited to resolving a short one-cycle timing interval at a
precision of a few parts in $10^4$. Frame rate alone is insufficient: exposure blur,
pixel calibration, lens distortion, and a poorly defined crossing threshold can
contribute more than the nominal frame interval.

Treat timestamp errors according to the acquisition method. A camera whose reported
frames share one clock has a nearly common timebase scale error across every frame;
adding cycles does not average that calibration error away. Frame-selection errors at
the beginning and end of one interval are more nearly independent. A photogate can
reduce spatial ambiguity, but its beam width and threshold location define a finite
event region. Verify that the bob clears the beam at the same height on every pass and
record the direction used for each trigger.

Video coordinates need a geometric calibration as well as a time calibration. Place a
scale in the swing plane, keep the optical axis approximately normal to that plane,
and correct lens distortion when the bob spans a large part of the image. A ruler in
front of or behind the swing plane gives an incorrect pixel-to-length conversion. The
camera must remain stationary after calibration; a shifted tripod changes the image
mapping even when the pendulum itself is unchanged.

Sampling density also constrains a fitted sinusoid. Several samples around each
equilibrium crossing constrain phase and frequency more effectively than sparse
samples near a turning point, where the coordinate changes slowly. A long record
improves frequency resolution only while amplitude, effective length, and support
condition remain sufficiently stable. Divide a long record into adjacent windows and
compare their fitted periods when slow drift is plausible. A systematic change across
windows belongs in the model-bias discussion rather than in a single pooled standard
deviation.

Linear interpolation between two samples straddling an equilibrium crossing can
improve a timestamp when the measured coordinate is approximately linear over that
short interval. Near equilibrium, angular speed is largest and the displacement
curve is locally close to a line, making interpolation well conditioned. Near a
turning point, speed approaches zero; position noise then produces a large timing
uncertainty. A video analysis should therefore use a consistent center crossing or
fit the sinusoidal trajectory over a full cycle, rather than infer a period from
turning-point frames alone.

Keep raw timestamps, frame indices, coordinate calibration, and the crossing rule
with the derived period. The record permits a revised fit or a frame-count audit
without re-running the apparatus, and separates a numerical-reduction change from a
physical-experiment change.

**Scope and transfer of the pendulum model.**

Any physical pendulum has the same finite-amplitude correction after replacing
$g/L$ by $MgD/I_P$ in the small-angle frequency. Its exact angular equation has the
form $\ddot\theta+\Omega_0^2\sin\theta=0$, with
$\Omega_0^2=MgD/I_P$. The small-angle physical-pendulum period is multiplied by the
same elliptic-integral correction factor at the same angular amplitude. Mass
distribution sets the base period, while the nonlinear sine term supplies the
amplitude correction.

The present analysis assumes a freely swinging, weakly dissipative pendulum with no
periodic external drive. Torsional pendula, mass--spring oscillators, driven response,
and coupled oscillators require different restoring relations or additional degrees
of freedom. Their period formulas cannot be substituted into the simple-pendulum
gravimetry equation. The local checks above—geometry, amplitude, planarity, event
definition, and residual behavior—supply the boundary between a valid pendulum
measurement and an apparatus that needs a richer model.