Equations of the Uniform Circular Motion (U.C.M.)

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 formulas that correspond to this type of motion
• The relationship between linear quantities and angular quantities
• The concept of the period and frequency in the u.c.m
• How to deduce the u.c.m. equations

The equations for the u.c.m.

Relationship between angular and linear quantities

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

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:

$$f=1/T$$

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:

$$\left.\begin{array}{c}{\omega }_a=\omega \\ {\omega }_a=\frac{\mathrm{\Delta }\phi }{\mathrm{\Delta }t}=\frac{\phi -{\phi }_0}{t-t_0}\underset{t_0=0}{\underbrace{=}}\frac{\phi -{\phi }_0}{t}\end{array}\right\}\to \phi -{\phi }_0=\omega \cdot t\to \boxed{\phi ={\phi }_0+\omega \cdot t}$$