Uniform Circular Motion: period, frequency and angular velocity.When we travel by car we sometimes face curves. Maybe we don’t realize it, but on those occasions we describe arcs of circles. This means that we move along a circular trajectory. The motions that describe arcs of circumference are called **circular motions** ; if the velocity remains constant in magnitude during the circular motion, we find ourselves in a particular case called **uniform circular motion** .

We note that the speed, as a vector quantity, *cannot remain constant along a curved trajectory* : in fact the speed always has a tangent direction to the trajectory; if the latter is curved, the tangent direction continues to change and consequently the speed, as a vector, does not remain constant. What can remain constant, however, is the *speed modulus* .

Let us now consider a material point animated by uniform circular motion. During this motion, due to the “constant velocity modulus” condition, the point describes equal circumferential arcs (movement performed) in equal time intervals (necessary to complete these movements). What characterizes uniform circular motion is therefore the **period** , i.e. the **time taken by the point to complete a complete revolution** . The **unit of measurement**** of the period** , this being an interval of time, is the **second** .

On the Ferris wheel of an amusement park for example, the period is the duration of the ride, from when we go up to when we get off.

**frequency**

We can talk about **frequency**, i.e. the **number of complete revolutions described in one second** . The **frequency** is obtained, by its definition, by calculating the **reciprocal of the period** , and is measured, in the International System,a unit known as **Hz (Hertz)**

## velocity

Since velocity (as a vector) takes the direction tangential to the trajectory, it is often referred to as **tangential velocity**. The modulus of the tangential velocity is always given by its definition, $Δx $; the distance traveled in one complete revolution is given by$2πr$, Where is the radius of the circle $π$is the measured lap angle, while the time taken to travel an entire lap is, by definition, the period $T$.

In addition to this, there is another speed called **angular speed,** : it is defined as **the ratio between the portion of the angle described by the material point and the time interval used to describe it** :$ω=t_ $Its unit of measurement is$rad/s$( radians per second), different from that used to measure tangential speed. Remembering the definition of period and frequency, we can reach the formulas$ω=rv =Tπ =2πf$As can be seen from the formulas, the angular velocity **does not depend on the ****radius of the circle** described by the motion.

The (tangential) **velocity** , during uniform circular motion, remains constant in magnitude, but **continues to change in direction** . Responsible for this change is **acceleration** : for the definition of acceleration , in fact, it is the ratio between a change in speed and the time taken to make this change.

It can be demonstrated that the acceleration present in uniform circular motion has a direction perpendicular to the tangent to the circumference, pointing towards the center of the latter: for this reason it is defined as **centripetal acceleration** . His form is valid to=�2$to_{c}=ω_{2}r=rv $

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