In uniform circular motion, an object moves around a circle of radius
\(r\)
at constant speed.
Although the speed does not change, the direction of the velocity changes continuously, so the motion is still accelerated.
The key idea is that the velocity is always tangent to the circle, while the required acceleration points inward toward the center.
Period and frequency
If the object completes
\(N\)
rotations in total time
\(t\),
then the period
\(T\)
is the time for one rotation:
\[
\begin{aligned}
T &= \frac{t}{N}
\end{aligned}
\]
The frequency
\(f\)
is the number of rotations per unit time:
\[
\begin{aligned}
f &= \frac{N}{t}
\end{aligned}
\]
These are reciprocals:
\[
\begin{aligned}
f &= \frac{1}{T}
\end{aligned}
\]
Angular displacement and angular velocity
One full rotation corresponds to an angle of
\(2\pi\)
radians.
So after
\(N\)
rotations, the total angular displacement is
\[
\begin{aligned}
\varphi &= N \cdot 2\pi
\end{aligned}
\]
Dividing by the total time gives the angular velocity
\(\omega\):
\[
\begin{aligned}
\omega &= \frac{\varphi}{t}
\end{aligned}
\]
Since
\(\varphi = 2\pi N\)
and
\(f = N/t\),
this can also be written as
\[
\begin{aligned}
\omega &= 2\pi f
\end{aligned}
\]
Tangential speed
The object moves along the circumference, so the speed along the path is called the
tangential speed.
It is related to the angular velocity by
\[
\begin{aligned}
v &= r\omega
\end{aligned}
\]
This makes sense dimensionally:
radius gives distance, and angular velocity gives the rate of turning.
Their product gives distance per unit time.
Centripetal acceleration
Even though the speed is constant, the velocity direction changes at every instant.
That direction change requires an inward acceleration called the
centripetal acceleration:
\[
\begin{aligned}
a_c &= \frac{v^2}{r}
\end{aligned}
\]
Using
\(v = r\omega\),
this can also be rewritten as
\[
\begin{aligned}
a_c &= r\omega^2
\end{aligned}
\]
These two formulas are completely equivalent.
The calculator shows both so you can verify the same result in two different ways.
Position and component motion
If the motion starts from the positive x-axis and proceeds counterclockwise, then the angular position at time
\(t\)
is
\[
\begin{aligned}
\theta(t) &= \omega t
\end{aligned}
\]
The Cartesian coordinates are
\[
\begin{aligned}
x(t) &= r\cos(\omega t) \\
y(t) &= r\sin(\omega t)
\end{aligned}
\]
The velocity components are
\[
\begin{aligned}
v_x(t) &= -v\sin(\omega t) \\
v_y(t) &= v\cos(\omega t)
\end{aligned}
\]
So the graph of the motion in time is sinusoidal in x and y, even though the path in space is circular.
This is why the upgraded calculator includes both a circle animation and time-based component graphs.
Worked example
Use the sample values
\(r = 2\,\text{m}\),
\(t = 10\,\text{s}\),
and
\(N = 5\)
rotations.
Step 1. Period
\[
\begin{aligned}
T &= \frac{t}{N} = \frac{10}{5} = 2\,\text{s}
\end{aligned}
\]
Step 2. Frequency
\[
\begin{aligned}
f &= \frac{N}{t} = \frac{5}{10} = 0.5\,\text{Hz}
\end{aligned}
\]
Step 3. Angular displacement
\[
\begin{aligned}
\varphi &= N \cdot 2\pi = 5 \cdot 2\pi = 10\pi\,\text{rad}
\end{aligned}
\]
Step 4. Angular velocity
\[
\begin{aligned}
\omega &= \frac{\varphi}{t} = \frac{10\pi}{10} = \pi\,\text{rad s}^{-1}
\end{aligned}
\]
Step 5. Tangential speed
\[
\begin{aligned}
v &= r\omega = 2\pi \approx 6.28\,\text{m s}^{-1}
\end{aligned}
\]
Step 6. Centripetal acceleration
\[
\begin{aligned}
a_c &= \frac{v^2}{r} = \frac{(6.28)^2}{2} \approx 19.74\,\text{m s}^{-2}
\end{aligned}
\]
The same centripetal acceleration also comes from
\(a_c = r\omega^2\),
which is a very useful consistency check.
Tangential versus centripetal
| Quantity |
Direction |
Role |
| Tangential velocity \(\vec v\) |
Tangent to the circle |
Shows the instantaneous direction of motion |
| Centripetal acceleration \(\vec a_c\) |
Toward the center |
Changes the direction of the velocity |
| Radius vector \(\vec r\) |
From center to object |
Locates the object on the circle |
The animation in the calculator makes this distinction visual.
The velocity vector remains tangent to the circle, while the centripetal-acceleration arrow always points inward.
The green and red arrows show the x- and y-components of the velocity, helping you connect the circular motion to the sinusoidal component graphs.
Summary
| Quantity |
Formula |
Meaning |
| Period |
\(T = t/N\) |
Time for one full rotation |
| Frequency |
\(f = N/t\) |
Rotations per unit time |
| Angular displacement |
\(\varphi = N \cdot 2\pi\) |
Total angle turned |
| Angular velocity |
\(\omega = \varphi/t = 2\pi f\) |
Turning rate |
| Tangential speed |
\(v = r\omega\) |
Linear speed along the path |
| Centripetal acceleration |
\(a_c = v^2/r = r\omega^2\) |
Inward acceleration needed to keep the object in circular motion |
