A body performs an uniform circular motion (u.c.m.) when its trajectory is a circumference and its angular velocity is constant. In this section, we are going to study:

## The equations for the u.c.m.

Uniform circular motion equations are as follows:

$\phi ={\phi }_{0}+\omega ·t$
$\omega =\text{constant}$
$\alpha =0$

Where:

• $\phi$, ${\phi }_{0}$: Angular position of the body at the time studied and at the initial moment, respectively. Its unit in the International System (S.I.) is the radian (rad)
• $\omega$: Angular velocity of the body. Its unit in the International System (S.I.) is the radian per second (rad/s)
• $\alpha$: Angular acceleration. Its unit in the International System (S.I.) is radian per second squared (rad/s2)
Experiment and Learn

Position in the u.c.m.

Let us visualize how the angular position of a body in a u.c.m is calculated, after a certain time has passed.

To do this, move the sliders of the initial angular position (that is, the angle from which you want to begin the motion) and the angular velocity (angles per second) to set these values. Remember that angles are always measured in radians (180° = π radians = 3.1416 radians).

Press the play button and automatically you will have control of the time in the motion. Drag the slider t, to move time forward or backward and you will see what is the new position of the body at the selected time.

## Relationship between angular and linear quantities

We can relate angular and linear quantities in circular motions using the radius R.

Linear quantities Relationship Angular quantities
Distance traveled (s)
$s=\phi ·R$
φ
Linear velocity (v)
$v=\omega ·R$
ω
Tangential acceleration (at)
${a}_{t}=\alpha ·R$
α
Normal acceleration (an)
${a}_{n}=\frac{{v}^{2}}{R}={\omega }^{2}·R$
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## Period and frequency in u.c.m.

Uniform circular motion (u.c.m.) is a periodic motion, that is, it is repeated a certain number of times with the same characteristics. This allows us to define the following magnitudes:

• Period: It is the time that it takes the body to complete a rotation. It is represented by T and is measured in seconds (s). Its expression is given by:

$T=2\mathrm{\pi }/\mathrm{\omega }$
• Frequency: It is the number of rotations per second that the body describes. It is represented by f and its unit is the inverse of second (s-1), which is also called Hertz (Hz). Its expression is given by:
$f=\frac{\omega }{2·\pi }$

The frequency is the reciprocal of the period. Relating frequency, period and angular velocity using the previous expressions, we get:

$f=1/T$

$\omega =\frac{2·\pi }{T}=2·\pi ·f$

Finally, remember that the relationship between angular velocity and lineal velocity allows us to write the last of our expressions which relates angular velocity, lineal velocity, period, frequency and radius in uniform circular motion (u.c.m.):

$v=\omega ·R=\frac{2·\mathrm{\pi }}{T}·R=2·\mathrm{\pi }·f·R$

Do not forget that the concept of frequency and period only makes sense in periodic motion, thus, it makes no sense to talk about frequency or period in uniformly accelerated circular motion, for example.

## Deduction of equations of the u.c.m.

To obtain the equations of the uniform circular motion (u.c.m.) let us proceed in way similar to the one we used in the uniform rectilinear motion (u.r.m.), but considering angular magnitudes, rather than linear. We will consider the following properties:

• Angular acceleration is zero ($\alpha =0$)
• On the other hand, this implies that the average and instantaneous angular velocity of the motion have always the same value

With these restrictions, we get: