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:

$\begin{array}{c}{\omega }_{a}=\omega \\ {\omega }_{a}=\frac{\mathrm{\Delta }\phi }{\mathrm{\Delta }t}=\frac{\phi -{\phi }_{0}}{t-{t}_{0}}\underset{{t}_{0}=0}{\underset{⏟}{=}}\frac{\phi -{\phi }_{0}}{t}\end{array}}\to \phi -{\phi }_{0}=\omega \cdot t\to \overline{)\phi ={\phi }_{0}+\omega \cdot t}$

## Solved exercises worksheet

Here you can test what you have learned in this section.

#### A train ride

difficulty

A toy train nicknamed "torpedo" moves in a circular trajectory of 2 m radius without possibility of changing its linear velocity. Knowing it takes 10 seconds for a full rotation, calculate:

a) Its angular and linear velocities.
b) The angle and the distance traveled in 2 minutes.
c) Its acceleration.

#### Normal acceleration in a wheel

difficulty

A 1 m radius circular disk rotates at constant angular velocity, so that it takes 1.2 s for a full rotation. What is the normal or centripetal acceleration of the external points of its periphery?

#### Centripetal acceleration of the Moon

difficulty

Determine the centripetal acceleration of the Moon knowing that a complete orbit around the Earth takes 27.32 days (sidereal period) and the average distance is 384000 km.

#### An encounter in u.c.m.

difficulty

Two bodies, c1 and c2, begin to move from the same point at constant angular velocity, but in opposite directions, along a 30 m radius circumference. If the first one takes 20 seconds to complete a rotation and the second takes 60 seconds, calculate:

a) The time that they take to meet.
b) The angle and distance traveled by each one.

## Formulas worksheet

Here is a full list of formulas for the section Equations of Uniform Circular Motion (U.C.M.). By understanding each equation, you will be able to solve any problem that you may encounter at this level.

Click on the icon   to export them to any compatible external program.

#### Angular position in u.c.m.

$\phi ={\phi }_{0}+\omega ·t$

#### Angular velocity in u.c.m.

$\omega =\text{constant}$

#### Angular acceleration (u.c.m.)

$\alpha =0$

#### Frequency in uniform circular motion (u.c.m.)

$f=\frac{\omega }{2·\pi }$

#### Relationship angular velocity - period - frequency in u.c.m.

$\omega =\frac{2·\pi }{T}=2·\pi ·f$