---
title: Circular Motion
module: Kinematics
moduleNumber: 1
lessonNumber: 5
order: 105
summary: >
  An object going around a circle at a steady speed is still accelerating, because
  its velocity is forever changing direction — the fact that governs everything from a
  car on a curve to a satellite in orbit. We tie the angular description (angle,
  angular velocity, angular acceleration) to the linear one through $v=r\omega$, split
  the acceleration into an inward part that turns the velocity and a tangential part
  that changes its speed, and extend the inward $v^2/r$ result to any curved path
  through its local radius of curvature. Constant angular acceleration then mirrors
  straight-line motion equation for equation.
topics: [Kinematics]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 3 — Motion in Two and Three Dimensions; §3-3"
---

## Circular kinematics and moving bases

A particle moving on a circle of radius $r$ has arc length and angular displacement
related by $s=r\theta$. Differentiation gives

$$
v=r\omega,
\qquad \omega=\frac{\d\theta}{\d t}.
$$

Velocity is tangent to the path. Even at constant speed, its direction changes;
the resulting radial acceleration has magnitude

$$
a_r=\frac{v^2}{r}=r\omega^2.
$$

The acceleration points inward. A tangential acceleration $a_t=\d v/\d t=r\alpha$
is present when speed changes, with $\alpha=\d\omega/\d t$.

$$
% caption: Tangential velocity is perpendicular to the radius, while radial
% acceleration points toward the center. The perpendicular arrows distinguish
% tangential acceleration, which changes speed, from radial acceleration, which
% changes direction.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \definecolor{accel}{HTML}{2E9BE0}
  \draw[black] (0,0) circle (1.9);
  \draw[fill=white, draw=black, thick] (0,0) circle (2pt) node[below left] {center};
  \draw[black, thick] (0,0) -- (1.09,1.556) node[midway, below right] {radius};
  \draw[fill=acc!14, draw=acc, thick] (1.09,1.556) circle (2.4pt);
  \draw[->, acc, very thick] (1.09,1.556) -- (0.19,2.19) node[above] {$v$};
  \draw[->, accel, very thick] (1.09,1.556) -- (0.574,0.819) node[midway, right] {$a_r$};
\end{tikzpicture}
$$

**Position, velocity, and the local moving basis.**

Two perpendicular directions describe the motion locally:

- **Radial:** from the centre to the particle.
- **Tangent:** along the instantaneous velocity.

At angular coordinate $\theta$, the position is

$$
\vec r=r(\cos\theta\,\hat\imath+\sin\theta\,\hat\jmath).
$$

The magnitude $r$ is constant, but the direction changes as the particle moves, so
the time derivative is nonzero. Differentiating gives a velocity perpendicular to
the radius,

$$
\vec v=r\dot\theta(-\sin\theta\,\hat\imath
+\cos\theta\,\hat\jmath)=r\omega\,\hat T.
$$

The signed angular speed $\omega$ sets the tangent direction. Reversing the sense of
travel reverses both $\omega$ and $\vec v$, while the inward radial acceleration
still points toward the centre. The radial and tangent directions rotate with the
particle, so neither is a fixed Cartesian axis over a full orbit.

## Inward acceleration and angular rates

Acceleration is the rate of change of the velocity vector. On a uniform circle the
velocity magnitude is equal at two closely spaced times, but the directions differ.
Placed tail to tail, the two velocity vectors have a difference that points inward
in the small-angle limit; dividing it by the elapsed time gives the radial
acceleration.

Similar triangles relate that difference to the angular advance $\Delta\theta$:

$$
|\Delta\vec v|\approx v\Delta\theta,
\qquad
\Delta s=r\Delta\theta=v\Delta t.
$$

Taking the ratio and then the limit yields

$$
a_r=\frac{|\Delta\vec v|}{\Delta t}
=\frac{v^2}{r}=r\omega^2.
$$

The inward direction is called centripetal, meaning centre-seeking. Tension,
gravity, static friction, a normal force, or an electric force can supply the radial
net force. Force analysis belongs to dynamics; the kinematic result needs only a
curved path and a changing velocity direction.

$$
% caption: Two equal-speed velocity vectors separated by a small arc. Their vector difference points toward the circle centre. As the angular separation shrinks, the direction of the velocity change becomes exactly inward, producing the radial acceleration v squared over r.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[thick] (0,0) circle (1.80);
  \draw[fill=white, draw=black, thick] (0,0) circle (2pt);
  \fill (1.48,.96) circle (2pt);
  \fill (1.20,1.34) circle (2pt);
  \draw[->,very thick] (1.48,.96) -- (.80,2.01);
  \draw[->,very thick] (1.20,1.34) -- (.22,2.11);
  \node at (1.38,2.62) {earlier v};
  \node at (-.26,2.48) {later v};
  \draw[->,acc,very thick] (1.48,.96) -- (.72,.47) node[midway, below right] {change};
  \draw[dashed] (0,0) -- (1.48,.96);
  \draw[dashed] (0,0) -- (1.20,1.34);
\end{tikzpicture}
$$

> **Worked example.** A car rounds a flat curve of radius $60.0\ \mathrm m$ at a constant speed of $25.0\ \mathrm{m\,s^{-1}}$. Find its centripetal acceleration.
>
> The speed is constant, so only the radial component is present:
> $$
> a_r=\frac{v^2}{r}=\frac{(25.0)^2}{60.0}=10.4\ \mathrm{m\,s^{-2}}.
> $$
> The inward acceleration is $10.4\ \mathrm{m\,s^{-2}}$, about $1.06\,g$, directed toward the centre of the curve.

**Period, frequency, and angular-speed units.**

One revolution corresponds to an angular displacement of $2\pi$ radians. The
period $T$ is the time for that revolution, and the frequency $f$ is the number of
revolutions per second:

$$
f=\frac1T,
\qquad
\omega=\frac{2\pi}{T}=2\pi f,
\qquad
v=\frac{2\pi r}{T}=2\pi rf.
$$

Radians carry no independent physical dimension, so angular velocity has unit
$\mathrm{rad\,s^{-1}}$, customarily written $\mathrm{s^{-1}}$ in dimensional
calculations. A rotation rate given in revolutions per minute requires conversion
before use with SI radii. If a wheel turns at $n$ revolutions per minute, then

$$
f=\frac{n}{60},
\qquad
\omega=\frac{2\pi n}{60}.
$$

Two scaling rules follow:

- Doubling the radius at fixed frequency doubles speed and radial acceleration.
- Doubling the frequency at fixed radius doubles speed but quadruples radial
  acceleration, since $a_r$ scales as $\omega^2$.

The square dependence on $\omega$ is why high-speed rotating equipment is limited by
radial loads even when its speed has risen only moderately.

> **Worked example.** A laboratory turntable of radius $0.180\ \mathrm m$ rotates at $45.0\ \mathrm{rev\,min^{-1}}$. Find its rim speed and radial acceleration.
>
> The frequency is $45.0/60=0.750\ \mathrm{Hz}$, so the angular speed is
> $$
> \omega=2\pi(0.750)=4.71\ \mathrm{rad\,s^{-1}}.
> $$
> Rim speed and radial acceleration follow:
> $$
> v=r\omega=(0.180)(4.71)=0.848\ \mathrm{m\,s^{-1}},
> \qquad
> a_r=r\omega^2=3.99\ \mathrm{m\,s^{-2}}.
> $$
> The rim speed is $0.848\ \mathrm{m\,s^{-1}}$ and the inward acceleration is $3.99\ \mathrm{m\,s^{-2}}$, less than $g$ but large enough that an object resting on the turntable needs substantial static friction to stay at the same radius.

## Nonuniform motion and curvature

When the speed changes, acceleration has two components. The tangent component

$$
a_t=\frac{\d v}{\d t}=r\alpha
$$

changes speed. The radial component $a_r=v^2/r$ changes direction. Their
perpendicularity gives the total magnitude

$$
a=\sqrt{a_t^2+a_r^2}.
$$

An accelerating bicycle on a circular track has both components: its velocity
grows along the track while its direction continues to turn inward. A constant
speed removes $a_t$ but leaves $a_r$. A particle that briefly stops has $a_r=0$
at that instant, even if a nonzero tangent acceleration immediately starts it
moving in the opposite direction. The formula $v^2/r$ therefore describes the
local speed-dependent component, not the total acceleration of every circular-path
motion.

$$
% caption: Nonuniform circular motion has two perpendicular acceleration components. The tangent component changes the speed, the inward component bends the velocity direction, and the diagonal vector is their instantaneous total acceleration.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black,thick] (0,0) circle (1.75);
  \draw[fill=white, draw=black, thick] (0,0) circle (2pt);
  \draw[fill=acc!12, draw=acc, thick] (1.40,1.05) circle (2.3pt);
  \draw[->,black,thick] (1.40,1.05) -- (.35,.25);
  \node[black] at (-.25,.12) {$a_r$};
  \draw[->,acc,thick] (1.40,1.05) -- (.58,2.15) node[above] {along path};
  \draw[->,black,very thick] (1.40,1.05) -- (-.05,1.35) node[left] {total a};
\end{tikzpicture}
$$

> **Worked example.** A particle on a circle of radius $2.40\ \mathrm m$ speeds up from $3.00$ to $5.00\ \mathrm{m\,s^{-1}}$ in $4.00\ \mathrm s$. Find the acceleration components and total magnitude at the later instant.
>
> The tangential component is the rate of change of speed:
> $$
> a_t=\frac{5.00-3.00}{4.00}=0.500\ \mathrm{m\,s^{-2}}.
> $$
> At $v=5.00\ \mathrm{m\,s^{-1}}$ the radial component and total magnitude are
> $$
> a_r=\frac{(5.00)^2}{2.40}=10.4\ \mathrm{m\,s^{-2}},
> \qquad
> a=\sqrt{a_t^2+a_r^2}=10.4\ \mathrm{m\,s^{-2}}.
> $$
> The total is $10.4\ \mathrm{m\,s^{-2}}$ to this precision. Here $a_t$ barely changes the magnitude, but it still fixes the changing speed and points perpendicular to the large radial component.

**Curvature beyond a perfect circle.**

The radial result extends to any smooth curved path. At each point the osculating
circle has radius of curvature $\rho$, and the normal acceleration is

$$
a_n=\frac{v^2}{\rho}.
$$

A circle has $\rho=r$ everywhere. On a road bend, roller-coaster hill, or curved
trajectory, $\rho$ changes along the path. A broad bend has large $\rho$ and a
smaller normal acceleration at the same speed. A sharp bend has small $\rho$ and a
larger normal acceleration. A straight segment has infinite radius of curvature,
so its normal acceleration is zero. This local construction separates turning of
the path from changes in speed along the path.

$$
% caption: A general smooth path has a local osculating circle at a selected point. The local radius sets the normal acceleration v squared over radius; a tighter bend produces a larger inward acceleration at the same speed.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[acc,very thick,domain=-2.6:2.45,samples=120]
    plot (\x,{0.15*\x+0.92*sin(48*\x+40)});
  \draw[black,dashed] (.08,-.55) circle (1.58);
  \draw[fill=acc!12, draw=acc, thick] (.20,1.05) circle (2.3pt);
  \draw[fill=white, draw=black, thick] (.08,-.55) circle (2pt);
  \draw[->,black,thick] (.20,1.05) -- (.08,-.55) node[midway,right] {radius};
  \draw[->,acc,thick] (.20,1.05) -- (1.15,1.16);
  \node[acc,anchor=west] at (1.38,1.55) {path direction};
  \node[below] at (.08,-.70) {local center};
\end{tikzpicture}
$$

A car can hold constant speed while its normal acceleration rises and falls as the
road tightens and opens. A satellite on an ellipse has continuously changing speed
and radius of curvature, so no single radius or single radial acceleration describes
the whole orbit. At any instant, though, the tangent and normal components give a
complete local decomposition.

## Parametric trajectories and instantaneous limits

A circular path is represented most directly by two time-dependent Cartesian
coordinates. With the centre at $(x_c,y_c)$, radius $r$, and angular coordinate
$\theta(t)$,

$$
x(t)=x_c+r\cos\theta(t),
\qquad
y(t)=y_c+r\sin\theta(t).
$$

The geometry holds when angular speed varies, and the sign convention is explicit:
increasing $\theta$ is counterclockwise motion in the standard $x$-$y$ plane. A shift
of the centre changes the position coordinates but not the velocity or acceleration
produced by the circular displacement. Reconstructing a point needs both the radius
(a length) and $\theta$ (an angle).

Differentiating with respect to time gives

$$
v_x=-r\dot\theta\sin\theta,
\qquad
v_y=r\dot\theta\cos\theta.
$$

The velocity magnitude is $r|\dot\theta|$, since the squared components add to
$r^2\dot\theta^2$. The velocity is perpendicular to the radius for every $\theta$:
the scalar product $(\vec r-\vec r_c)\cdot\vec v$ vanishes identically. Changing the
sign of $\dot\theta$ reverses the velocity without changing its speed.

A second differentiation separates the acceleration into two terms:

$$
\vec a=
-r\dot\theta^2(\cos\theta\,\hat\imath+\sin\theta\,\hat\jmath)
+r\ddot\theta(-\sin\theta\,\hat\imath+\cos\theta\,\hat\jmath).
$$

The first term points toward the centre and has magnitude $r\omega^2$. The second
term is tangent to the path and has magnitude $r|\alpha|$. The signs are important.
A negative angular acceleration does not mean an outward acceleration; it means that
the tangent component points opposite the direction defined by increasing angle.
The radial term remains inward for either sense of rotation because it depends on
$\omega^2$.

For uniform circular motion, $\theta(t)=\theta_0+\omega t$. Substitution into the
parametric equations yields a sinusoidal record for each Cartesian coordinate, even
though the physical trajectory is a circle. The coordinate oscillation has period
$2\pi/|\omega|$, while the two coordinates are separated by one-quarter of a cycle.
A position sensor that reports only $x(t)$ cannot by itself distinguish a particle
on a circle from another system with the same sinusoidal horizontal motion; the
second coordinate or a geometric constraint identifies the path.

**Finite angular intervals and instantaneous limits.**

An angular displacement over a measured interval is finite:

$$
\Delta\theta=\theta(t+\Delta t)-\theta(t).
$$

It determines an average angular velocity
$\omega_{\rm av}=\Delta\theta/\Delta t$ and an arc displacement
$\Delta s=r\Delta\theta$ only when the path radius is constant. The average velocity
vector, however, is displacement divided by time and points along the chord joining
the two positions. It is not generally tangent to either endpoint. Confusing arc
length divided by time with the magnitude of average velocity hides this directional
difference.

At finite angular advance, the straight-line displacement magnitude is

$$
|\Delta\vec r|=2r\sin\left(\frac{|\Delta\theta|}{2}\right),
$$

whereas the arc length is $r|\Delta\theta|$. The two agree only in the small-angle
limit. Their ratio is $2\sin(|\Delta\theta|/2)/|\Delta\theta|$, which approaches one
as $\Delta\theta$ approaches zero. For a half revolution, the displacement is
$2r$ while the travelled distance is $\pi r$; treating them as interchangeable
would substantially underestimate the average speed along the path.

The instantaneous angular velocity is the derivative
$\omega=\lim_{\Delta t\to0}\Delta\theta/\Delta t$. The velocity is likewise
$\vec v=\lim_{\Delta t\to0}\Delta\vec r/\Delta t$. The chord direction approaches
the tangent as the interval shrinks, and the magnitude approaches $r|\omega|$. A
derivative is not a short interval declared to be zero; it is the limiting value
approached by a sequence of finite measurements or a smooth model fitted to them.

Finite intervals have direct measurement roles. A turntable tachometer reports an
average over several revolutions, a camera reports positions at discrete frames, and
a road survey reports curvature over a finite station interval. The interval must be
short enough that the desired local quantities do not change appreciably, yet long
enough that timing and position noise do not dominate the difference. That trade-off
is an experimental issue, not a change in the circular-motion equations.

## Reconstructing measured angular motion

Video tracking and rotary encoders provide positions at sampled times
$t_k$. For a known circle centre, each sample gives an angle through
$\theta_k=\atanTwo(y_k-y_c,x_k-x_c)$. The two-argument angle is needed
because an ordinary inverse tangent cannot identify the quadrant. The reported
angles must then be unwrapped: a sequence passing from $179^\circ$ to
$-179^\circ$ represents a small positive advance through $180^\circ$, not a
negative jump of $358^\circ$.

An interval estimate of angular speed is

$$
\omega_{k+1/2}\approx\frac{\theta_{k+1}-\theta_k}{t_{k+1}-t_k}.
$$

A centred estimate at an interior sample is usually more accurate for smooth motion:

$$
\omega_k\approx\frac{\theta_{k+1}-\theta_{k-1}}{t_{k+1}-t_{k-1}},
\qquad
\alpha_k\approx
\frac{\omega_{k+1/2}-\omega_{k-1/2}}{(t_{k+1}-t_{k-1})/2}.
$$

Finite differences approximate instantaneous derivatives over finite sample intervals.
Their reliability depends on frame rate, angle resolution, and the smoothness of
the motion. Differentiation amplifies random position noise, so angular acceleration
is normally noisier than angle or angular speed. A fitted trajectory, smoothing
method, or independent encoder can be preferable when acceleration is the quantity
of interest.

The graph fragment in the figure rises smoothly because the samples advance in one
sense around the circle. A reversal would produce a decreasing unwrapped angle and
a negative angular velocity. Missing samples create a further ambiguity: if the
object advances by more than $\pi$ radians between frames, the observed positions
alone cannot determine how many full turns occurred. The sampling rate must be high
enough that the expected angular advance between samples remains below the chosen
unwrapping threshold.

A reconstruction should be checked against the geometric constraint. The distances
$\sqrt{(x_k-x_c)^2+(y_k-y_c)^2}$ should remain near the stated radius; systematic
variation indicates an eccentric path, a mislocated centre, perspective distortion,
or tracking error. Once $\theta(t)$ has been reconstructed, the predicted tangent
speed $r|\omega|$ and radial acceleration $r\omega^2$ can be compared with
Cartesian velocity and acceleration estimates. Agreement tests both the circular
path model and the time differentiation, while disagreement identifies which
assumption or measurement needs revision.


Sampling also determines the distinction between angular position and accumulated
turn count. An encoder may report a wrapped angle in the interval from zero to one
revolution, while a motor controller requires the total angle after many turns.
Adding or subtracting whole $2\pi$ increments during unwrapping creates this
continuous accumulated coordinate. The choice is not cosmetic: average angular
speed over ten revolutions must use the total angular displacement, not the small
difference between the wrapped endpoints.

A practical data reduction begins by calibrating the centre and pixel or encoder
scale, then plotting radius and unwrapped angle against time before differentiating.
A radius plot that drifts with viewing angle can identify perspective error before it
appears as a fictitious angular acceleration. Repeated trajectories distinguish
random frame-to-frame scatter from repeatable speed variation. Averaging equivalent
phase samples reduces random scatter; a repeatable variation belongs in the model.
Timestamp irregularity must be retained in every finite difference; replacing
unequal frame intervals by a nominal frame rate biases the inferred angular speed.

When angular speed changes abruptly, a centred difference spreads the change over
neighbouring samples. This is a limitation of the estimator, not evidence that the
physical acceleration was gradual. The sample interval, differencing rule, and any
smoothing operation should therefore accompany a reported acceleration value. Those
details specify which finite-time quantity was measured and how closely it can stand
for an instantaneous circular-motion variable.

**Relative rotation, limits, and measured circular motion.**

Circular motion is usually described in a laboratory frame, but many questions
concern one rotating object relative to another. Angular differences are simple only
after fixing a common centre and sign convention, and a moving camera, a rotating
platform, or an encoder on a turntable records data in a frame that may not be
inertial. Kinematic variables keep their definitions in every frame; their
interpretation must match the reference motion.

**Relative angular coordinates.**

For two points moving about the same centre, their angular separation is

$$
\Delta\theta_{BA}=\theta_B-\theta_A,
\qquad
\omega_{BA}=\omega_B-\omega_A,
\qquad
\alpha_{BA}=\alpha_B-\alpha_A.
$$

Both quantities are signed. If both points rotate counterclockwise but point B has
the smaller angular speed, then B moves backward relative to A even while both have
positive laboratory angular velocity. A zero relative angular speed means that the
angular separation is fixed. It does not require the points to be stationary; two
markers on a rigid disk share one angular speed and maintain a fixed separation
while each has a nonzero tangential speed.

The encounter condition follows from the accumulated relative angle. Two marks on
the same circular track coincide whenever their relative angle changes by an integer
multiple of $2\pi$. For constant angular speeds, the time between successive
encounters is $2\pi/|\omega_B-\omega_A|$. This relation is valid for an observer who
uses the same centre and follows the same orientation convention for both marks.
Using one clockwise angle and one counterclockwise angle without retaining their
signs gives an incorrect relative speed.

Relative angle is not the same as relative distance. If two riders occupy different
radii on a carousel, their straight-line separation changes according to the vector
difference of their position vectors, even when their relative angle is constant.
Their tangential speeds are $r_A\omega_A$ and $r_B\omega_B$, so equal angular speed
does not imply equal linear speed. A rigid disk therefore has one angular velocity
but a continuum of tangential speeds and radial accelerations, each proportional to
radius.

A rotating observer adds a frame term. For a vector $\vec q$ measured from the
rotating observer's origin,

$$
\left(\frac{\d\vec q}{\d t}\right)_{\rm lab}
=
\left(\frac{\d\vec q}{\d t}\right)_{\rm rot}
+\vec\Omega\times\vec q.
$$

The cross-product term is the velocity associated with the observer's rotating
axes. A mark fixed on a disk has zero velocity in the disk frame but nonzero
laboratory velocity. When relative accelerations are required in a rotating frame,
further terms involving angular acceleration and Coriolis effects arise. Those are
frame-kinematics effects, not additional contact forces. For ordinary circular-path
problems, retaining the laboratory frame avoids them; for turntable measurements,
the rotation rate of the measurement frame must be stated.

**Average quantities, local derivatives, and constant angular acceleration.**

Average angular variables summarize a finite interval, while instantaneous variables
describe a local limit. The average angular velocity and average angular acceleration
are

$$
\omega_{\rm av}=\frac{\Delta\theta}{\Delta t},
\qquad
\alpha_{\rm av}=\frac{\Delta\omega}{\Delta t}.
$$

Endpoint differences determine them, but they do not specify how angle or angular
speed varied between those endpoints. A wheel can start and finish with
the same angular velocity while undergoing large positive and negative angular
accelerations in between. An average angular acceleration of zero only states that
the endpoint angular velocities match.

The instantaneous values are slopes of the time graphs:
$\omega=\d\theta/\d t$ and $\alpha=\d\omega/\d t$. A tangent line to a smooth
angle-time curve gives the instantaneous angular velocity; a secant joining two
separated samples gives an average. The two slopes coincide only when the curve is locally
linear over that interval. As the interval contracts, the secant approaches the
tangent, which is the geometric content of the derivative limit.

Constant angular acceleration is a restricted model. Only when $\alpha$
is constant over the stated interval may the linear-motion analogues be used:

$$
\omega_f=\omega_i+\alpha\Delta t,
\qquad
\Delta\theta=\omega_i\Delta t+\frac12\alpha(\Delta t)^2,
\qquad
\omega_f^2=\omega_i^2+2\alpha\Delta\theta.
$$

The last relation eliminates time without establishing constant acceleration. A
speed-control system that changes torque with angle or time generally has variable
angular acceleration. Apply the constant-acceleration equations only with a model or
measurement supporting that condition; otherwise integrate a known $\omega(t)$ or
infer local slopes from the data.

Average tangential acceleration is $r\alpha_{\rm av}$ only for a point at fixed
radius. For a deforming pulley, an extending radius, or a point that slips relative
to the wheel, the simple circular relation must be reconsidered. The distinction
between finite and instantaneous intervals also applies to radial acceleration:
$r\omega_{\rm av}^2$ is not generally the time average of $r\omega(t)^2$. Squaring
and averaging do not commute when angular speed varies.

**Speed limits set by normal acceleration.**

A specified normal-acceleration limit gives a purely kinematic speed envelope. If
the allowable inward acceleration magnitude is $a_{\max}$ at a circular radius $r$,

$$
v\le\sqrt{a_{\max}r},
\qquad
|\omega|\le\sqrt{\frac{a_{\max}}{r}}.
$$

The first expression shows that a larger-radius curve permits a higher linear speed
for the same acceleration limit. The second shows that the permitted angular speed
is lower for a larger circle. These statements are consistent because the same
linear speed corresponds to fewer radians per second on a larger radius. They apply
to a passenger-comfort limit, a bearing-load limit, or an instrument range before a
particular force mechanism is selected.

> **Worked example.** A highway ramp of radius $120\ \mathrm m$ is to keep the inward acceleration below a passenger-comfort limit of $3.0\ \mathrm{m\,s^{-2}}$. Find the maximum speed.
>
> The kinematic envelope is $v\le\sqrt{a_{\max}r}$:
> $$
> v_{\max}=\sqrt{(3.0)(120)}=19\ \mathrm{m\,s^{-1}}.
> $$
> The speed limit is about $19\ \mathrm{m\,s^{-1}}$ ($68\ \mathrm{km\,h^{-1}}$), set purely by the geometry and the acceleration cap, before any friction or banking model.

$$
% caption: A normal-acceleration limit creates a speed envelope. The curve gives
% the maximum permitted tangential speed as radius changes; points below it satisfy
% the stated kinematic limit, while points above it require more inward acceleration
% than the limit allows.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.45,0) node[right] {radius};
  \draw[->, black] (0,0) -- (0,3.55) node[above] {speed};
  \draw[acc, very thick, domain=0.45:5.85, samples=100]
    plot (\x,{0.05+1.03*sqrt(\x)});
  \fill[acc!10] (0.45,0) -- (0.45,0.62) -- (1.20,1.03) -- (2.10,1.43) -- (3.15,1.83) -- (4.35,2.18) -- (5.85,2.55) -- (5.85,0) -- cycle;
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  \draw[dashed, black] (3.15,1.22) -- (3.15,0);
\end{tikzpicture}
$$

The curve in the figure follows a square-root dependence, so it rises with radius
but with decreasing slope. A vehicle that doubles its bend radius can increase speed
only by a factor of $\sqrt2$ at the same normal-acceleration limit, not by a factor
of two. Conversely, doubling speed requires four times the radius to maintain the
same inward acceleration. These scaling checks are often more reliable than
memorizing individual numerical results.

The radius-speed relation is a local kinematic requirement. The force mechanism is
specified only after the available inward interaction has been modelled.

| Fixed quantity | Scaling consequence | Physical interpretation |
| --- | --- | --- |
| $a_n=v^2/r$ | $v\propto\sqrt r$ | larger radius permits higher speed at the same normal acceleration |
| $a_n$ and $v$ | $r\propto v^2$ | doubling speed requires four times the radius |
| $r$ and available force | $v_{\max}=\sqrt{Fr/m}$ | dynamics sets the limiting force |

A kinematic acceleration limit is not automatically a force limit. On a level road,
available static friction may set the maximum normal force; on a banked track,
normal force components contribute; on a rotating machine, material stress or a
bearing specification may dominate. The equation $v^2/r$ determines the required
inward acceleration. Dynamics determines whether the available interactions can
supply it. Separating these steps prevents a force assumption from being smuggled
into a kinematics calculation.

**Video reconstruction and experimental resolution.**

A video reconstruction must establish a physical coordinate system before angular
variables are differentiated. Camera pixels are not lengths until a scale is set in
the plane of motion. A camera aimed obliquely at a circular track projects the circle
to an ellipse, so the apparent centre and apparent radius vary with direction.
Rectifying the image with a planar calibration target, or aligning the optical axis
normal to the motion plane, is necessary before a pixel trajectory is treated as a
circle.

Frame times require the same care. Many cameras use variable frame timing, dropped
frames, exposure intervals, or rolling shutters. Each angular difference requires
the timestamp of its tracked image. A nominal frames-per-second value cannot resolve
variable timing. A short exposure can blur a fast marker across a non-negligible angular
arc; the recorded position may approximate a time average rather than the
instantaneous position at the frame label. These effects set a resolution floor for
angular speed and acceleration.

Repeated measurements support an uncertainty estimate. A single tracked point may
wander because of pixel selection, lens distortion, or occlusion. Tracking several
visible markers on a rigid disk permits a common-centre fit and separates rigid
rotation from individual marker error. The residual radius, the residual angular
fit, and the disagreement between independent markers give quantitative diagnostics.
A claim of constant angular speed requires residuals consistent with timing and
position uncertainty. A visually smooth line alone does not establish the model.

Rapidly rotating objects introduce a separate aliasing risk. If successive frames
cannot distinguish one marker advance from another differing by a full revolution,
the unwrapped angle is ambiguous. A distinctive multi-mark pattern, a higher frame
rate, a separate encoder, or a known maximum speed resolves the ambiguity. The
Nyquist-style requirement is practical: sample often enough that the largest expected
angular advance per frame is safely below the identification range of the marker
pattern.

The final reconstruction can be tested in both Cartesian and angular forms. Fit a
circle to the positions, obtain $\theta(t)$, and predict
$\vec v=r\omega\,\hat T$ and
$\vec a=-r\omega^2\,\hat r+r\alpha\,\hat T$. Independently estimate
Cartesian velocity and acceleration from the position sequence. Agreement is
consistent with the circular-path model. Systematic disagreement indicates an
eccentric track, changing radius, a misidentified centre, a nonplanar path, or an
inadequate sampling rate. The comparison turns circular-motion equations into a
measurement test of the trajectory model.

The reconstruction has one geometric fit and two derivative estimates. Keeping their
time labels together prevents a radius measured over one interval from being paired
with an angular speed inferred over another.

| Record | Derived quantity | Consistency test |
| --- | --- | --- |
| Calibrated positions | fitted centre and radius | residual radius versus angle |
| Unwrapped angle | $\omega=\d\theta/\d t$ | period or encoder comparison |
| Angular-rate change | $\alpha=\d\omega/\d t$ | tangent acceleration direction |
| Cartesian differences | $\vec v$ and $\vec a$ | agreement with $r\omega$ and $r\omega^2$ |


Relative-motion measurements require the same calibration discipline. When two
markers are tracked from a camera fixed to the laboratory, compute each unwrapped
laboratory angle first and then subtract them. Tracking only the visible separation
can fail near overlap, and a wrapped difference can create a false sudden reversal.
When the camera itself rotates with the platform, its measured angular rate must be
subtracted from both laboratory rates before an encounter time or relative speed is
reported.

An independent internal check compares the inferred period with separate timing. For
nearly uniform motion, the time for a marker to return to one reference direction
should match $2\pi/|\omega|$ from the unwrapped-angle slope. The mismatch should be
no larger than the combined uncertainty from timing, centre placement, and marker
identification. For nonuniform motion, integrate the fitted angular speed over the
observed interval and compare the result with the measured unwrapped angular advance.
The closure check detects lost turns and branch-cut errors that a local velocity
plot may not resolve.

Acceleration estimates deserve the strictest reporting because they involve two
numerical derivatives. A value of $r\omega^2$ is reliable only at the time associated
with the angular-speed estimate and only if the fitted radius represents that same
instant. If the radius varies, the normal component is $v^2/\rho$ with the local
radius of curvature rather than a nominal track radius. Recording the estimation
interval beside each reported speed and acceleration makes clear whether the result
is an average, a centred approximation, or a model-derived instantaneous value.

## Three-dimensional paths and rotating frames

Circular-path kinematics extends beyond a flat track. A particle can trace a
horizontal circle at a fixed height, move in a vertical plane, or be described from
a coordinate system that rotates with the apparatus. In every case, the local
tangent-normal decomposition remains valid: speed is tangent to the path and the
normal component of acceleration points toward the local centre of curvature. What
changes is the orientation of those local directions in space and the frame used to
describe them.

**Conical circular paths.**

A point attached to a line of fixed length $L$ can travel around a vertical axis at
a constant inclination $\beta$ from the vertical. Its path is a horizontal circle,
not a circle drawn in the plane of the line. The radius and height are set by the
geometry:

$$
r=L\sin\beta,
\qquad
z=z_{\rm pivot}-L\cos\beta.
$$

At constant angular speed about the vertical axis, the tangential velocity is
horizontal and has magnitude $v=r\omega$. The normal acceleration is also horizontal
and points from the moving point toward the rotation axis. It is not directed along
the supporting line unless the line happens to have the required orientation. A
side-view sketch can conceal this distinction by showing the line and radius in one
plane while the instantaneous velocity points perpendicular to that plane.

$$
% caption: A conical path has a fixed vertical height and a horizontal circular
% radius r. The blue arrow is a projected cue for tangent velocity around that
% horizontal circle, while normal acceleration points horizontally toward the axis;
% the string or rod geometry determines the radius through its inclination angle.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->] (0,-0.30) -- (0,3.55) node[above right] {axis};
  \draw[fill=white, draw=black, thick] (0,3.05) circle (2pt) node[above left] {pivot};
  \draw[thick] (0,3.05) -- (2.05,1.10);
  \draw[thick] (0,1.10) ellipse (2.05 and 0.42);
  \draw[dashed] (0,1.10) -- (2.05,1.10) node[midway, below] {$r$};
  \fill (2.05,1.10) circle (2.6pt);
  \draw[->, acc, very thick] (2.05,1.10) -- (2.05,2.18) node[right] {$v$};
  \draw[->, thick] (2.05,1.10) -- (0.70,1.10) node[midway, above] {$a_r$};
  \draw (0,2.35) arc (-90:-43:0.70);
\end{tikzpicture}
$$

The direction of motion around a horizontal circle can be reconstructed from a top
view, whereas the radius and inclination are easiest to measure from a side view.
Combining those views avoids treating the conical path as a planar pendulum arc.
Under nonuniform rotation, the tangential acceleration is horizontal and tangent to
the horizontal circle, while the normal component remains horizontal toward the
axis. A vertical component of acceleration would indicate changing height or a
changing cone angle. A fixed cone angle has no vertical acceleration component from
the circular-path kinematics alone.

The relation between period and radius has the same form as for any circle:
$T=2\pi r/v$. Reducing the angle $\beta$ reduces the radius and, at fixed angular
speed, reduces both tangential speed and normal acceleration. At fixed tangential
speed, reducing radius increases normal acceleration. These statements are
kinematic; a force model is needed to determine which speeds and inclinations a
string, track, or rotating mechanism can actually sustain.

**Vertical circular paths and speed extrema.**

A vertical circular path has a fixed geometrical centre, but its tangent and normal
directions sweep through every orientation. At the top and bottom of the circle,
the tangent is horizontal and the normal direction is vertical. At the side points,
the tangent is vertical and the normal direction is horizontal. The normal
acceleration always has magnitude $v^2/r$ and points inward; it does not switch from
inward to outward at the top of the path. What changes is its direction relative to
the laboratory vertical.

Speed extrema are identified kinematically by the tangential acceleration. Since
$a_t=\d v/\d t$, a smooth local maximum or minimum of speed occurs where $a_t=0$ and
changes sign. The radial component may be nonzero at the same instant.
At the top or bottom of a vertical circle, a nonzero radial acceleration therefore
does not imply that the speed is changing; it only indicates continuing change of
velocity direction. This is an important distinction in video analysis, where a
marker can be at a highest point with horizontal velocity and a substantial inward
acceleration.

$$
% caption: A vertical circle has inward normal acceleration at every point. At the
% top and bottom the tangent directions are horizontal, while the normal arrows are
% vertical toward the centre. A speed extremum is determined by a zero tangential
% component, not by disappearance of the inward normal component.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, thick] (0,0) circle (1.75);
  \draw[fill=white, draw=black, thick] (0,0) circle (2pt);
  \draw[fill=acc!12, draw=acc, thick] (0,1.75) circle (2.5pt);
  \draw[fill=acc!12, draw=acc, thick] (0,-1.75) circle (2.5pt);
  \draw[->, acc, very thick] (0,1.75) -- (1.20,1.75) node[right] {$v$};
  \draw[->, black, thick] (0,1.75) -- (0,0.55) node[midway, right] {$a_r$};
  \draw[->, acc, very thick] (0,-1.75) -- (-1.20,-1.75) node[left] {$v$};
  \draw[->, black, thick] (0,-1.75) -- (0,-0.55) node[midway, right] {$a_r$};
\end{tikzpicture}
$$

With known height-dependent speed, extrema can be found directly from the measured
or modelled function $v(\theta)$. The angular locations where
$\d v/\d\theta=0$ are also speed extrema provided angular speed is nonzero there,
because $\d v/\d t=(\d v/\d\theta)\,\omega$. A reversal point with
$\omega=0$ requires separate care: the speed is zero, radial acceleration vanishes,
and the tangent direction may reverse as motion restarts. Differentiating a noisy
speed record near such a point is unreliable, so the local geometry and position
sequence should be checked together.

A common energy model for a frictionless vertical path predicts different top and
bottom speeds because gravitational potential changes. That prediction belongs to
energy and dynamics, not to the circular-path identity $a_n=v^2/r$. Kinematics can
state the local normal acceleration once a speed is measured; it cannot infer the
speed change from height without an additional physical model. Keeping this boundary
clear prevents gravitational assumptions from being inserted into an otherwise
geometric calculation.

**The centrifugal description and its limits.**

An observer fixed to a platform rotating at constant angular velocity
$\vec\Omega$ uses noninertial coordinates. A stationary object at position
$\vec r$ in that rotating frame has laboratory acceleration
$\vec\Omega\times(\vec\Omega\times\vec r)$, directed toward the axis. A Newton-style
force balance in the rotating frame introduces an apparent centrifugal term:

$$
\vec a_{\rm cf}=-\vec\Omega\times
(\vec\Omega\times\vec r).
$$

A point perpendicular to the rotation axis has this apparent acceleration directed
outward with magnitude $\Omega^2r$. The inertial laboratory frame contains the
real inward acceleration required by the curved path. Both descriptions predict the
same measured trajectory when their frame-specific terms are used consistently.

The centrifugal term accounts for a scale reading on a rotating platform or for a
fluid surface observed from the rotating container. It becomes misleading when it
is imported into an inertial free-body diagram and added to real forces without
changing frames. A body that is released from a rotating disk continues according to
its laboratory velocity; it does not receive an outward impulse at release. The
outward curvature seen by a disk observer arises because the disk axes turn beneath
the freely moving object.

If the rotating observer has angular acceleration, a further Euler term appears. If
an object moves relative to the rotating frame, a Coriolis term appears. Omitting
these terms while retaining only centrifugal acceleration is inconsistent except
for the restricted case of a point fixed in a uniformly rotating frame. The term
“centrifugal force” should therefore always be accompanied by its frame and its
scope. Inertial-frame circular kinematics needs no such apparent term.

**Uncertainty checks for circular-path results.**

Circular-motion quantities amplify different measurement errors in different ways.
A radius error affects tangential speed linearly through $v=r\omega$ and normal
acceleration linearly through $a_n=r\omega^2$ when angular speed is treated as
independent. If speed rather than angular speed is measured directly,
$a_n=v^2/r$, so a fractional speed error is doubled in the acceleration estimate
while a fractional radius error enters with opposite sign. These scaling relations
identify which measurement deserves the tightest calibration.

For small independent uncertainties, a practical estimate for the normal
acceleration uncertainty from direct speed and radius measurements is

$$
\left(\frac{\sigma_{a_n}}{a_n}\right)^2
\approx
\left(2\frac{\sigma_v}{v}\right)^2+
\left(\frac{\sigma_r}{r}\right)^2.
$$

The approximation assumes random, uncorrelated errors and a local linear response.
A fitted radius and angular speed from the same video data are often correlated, so
a full fit covariance or repeated-trial spread is more defensible. Reporting many
digits from a high-frame-rate video does not reduce systematic centre or lens error.

Several internal checks require no new equipment. The fitted radius should be
constant within uncertainty; tangent velocity should be perpendicular to the fitted
radius; and the inward acceleration component from Cartesian finite differences
should agree with $v^2/r$ within the derivative noise. The sign of the tangent
component should agree with the slope of unwrapped angle. A failure of one check is
diagnostic: radial drift suggests geometry or calibration error, nonperpendicular
velocity suggests tracking or timing error, and a normal-acceleration mismatch can
indicate an eccentric or nonplanar path.

A final report benefits from separating measured facts from inferred quantities.
State the coordinate frame, centre estimate, radius method, time base, sample
interval, and differentiation or fit procedure. Then give angular position, angular
speed, tangent speed, and normal acceleration with an uncertainty or resolution
scale appropriate to the data. This record distinguishes a justified instantaneous
estimate from a long-interval average and supports an explicit test of the
circular-path assumption.

A conical or vertical path also requires a three-dimensional geometry check. A
side-view radius inferred from a projected image can differ from the true
horizontal radius; a vertical-circle recording can appear elliptical when the camera
is not normal to the plane. Multiple calibrated views or a known spatial reference
resolves the ambiguity. The local tangent-normal equations remain correct, but
their components must be projected into the same physical coordinates as the data.


A speed-extremum analysis can be checked without assuming a particular force law.
Construct a signed tangent coordinate along the path, estimate the speed from
successive arc positions, and compare neighbouring values with the local tangent
acceleration. At a smooth maximum, the speed values decrease on both sides and the
tangent acceleration changes from positive to negative; at a smooth minimum, the
opposite sign change occurs. The normal acceleration can remain large throughout
this interval. This method applies equally to a vertical loop, a conical track with
changing rotation rate, or a curved path reconstructed from video.

The physical meaning of an acceleration limit depends on the frame in which it is
reported. An accelerometer fixed to a rotating platform records proper acceleration
associated with the real support interactions, while a rotating-coordinate analysis
may additionally quote the apparent centrifugal field. A passenger's laboratory
trajectory, a platform-frame scale reading, and a camera attached to the platform
are related observations but should not be combined as if they were measurements of
one identical vector. Stating the sensor orientation and frame prevents an apparent
disagreement from becoming a false violation of circular kinematics.

For numerical work, retain units that expose the model. Radians are dimensionless
in differentiation, but writing radian per second beside angular speed and metre per
second squared beside normal acceleration makes a missed conversion from revolutions
or degrees easier to find. Convert an angle recorded in degrees before using
trigonometric derivatives or $r\omega$ relations. A final order-of-magnitude check
should compare the inferred period, tangent speed, radius, and normal acceleration:
$2\pi r/T$, $r\omega$, and $v^2/r$ must describe the same local motion within the
measurement uncertainty.

When those three estimates disagree beyond the stated resolution, the remedy is to
revisit the coordinate calibration and time record before attributing the mismatch
to an unfamiliar force or a failure of circular-path kinematics.

One measurement table keeps the geometric fit, angular fit, and derivative estimate
from being reported as unrelated numbers. The relations should refer to the same
time interval and the same fitted centre.

| Quantity | Estimate | Independent comparison |
| --- | --- | --- |
| Radius | circle fit to tracked positions | residual radius over the record |
| Angular speed | slope of unwrapped $\theta(t)$ | return time $2\pi/\lvert\omega\rvert$ |
| Tangent speed | $v=r\omega$ | Cartesian finite-difference speed |
| Normal acceleration | $a_n=v^2/r$ | inward Cartesian acceleration component |

[^tipler33]: Tipler and Mosca, _Physics for Scientists and Engineers_, 6th ed., §3-3.