Question

Fan is rotating at $120rpm$ , find its frequency and angular speed ( $\omega$ ).

Answer 1

In physics, frequency is defined as the number of waves that pass through a given location in one unit of time. Frequency is also defined as the number of periods or cycles per second. The hertz is the SI unit of frequency (Hz). One hertz equals one cycle in every second ( ${s^{ - 1}}$ ). The angular speed of a spinning body is the rate at which its central angle varies over time.
We have a relationship that is, the relation between frequency and time and between frequency and angular speed.
$f = \dfrac{n}{t}$ ………(1)
$f =$ frequency;
$n =$ the number of cycles.
$t =$ time.
and $\omega = 2\pi \times f$ ………(2)
where $\omega =$ angular speed.

Complete Step By Step Answer:
Here we have asked to find the frequency and angular speed of the fan rotating at a speed of $120$ rotations per minute (rpm). As we know frequency is expressed in unit time, that is in seconds. So we want to convert rotations per minute to rotation per second.
$120rpm \Rightarrow \dfrac{{120}}{{60}}$ rotations per second (rps), which exactly gives frequency. Hence frequency is,
$f = \dfrac{{120}}{{60}} = 2{s^{ - 1}} = 2Hz$ …………(3),
where $s$ = time in seconds and $Hz =$ Hertz (SI unit of frequency)
Angular velocity, also known as rotational velocity or angular frequency vector, is a vector measure of rotation rate that relates to how rapidly an item spins or revolves relative to another point, i.e. how quickly an object's angular location or orientation changes with time. The magnitude of angular velocity is angular speed, a scalar quantity.
It has an angle per unit time dimension. Radians per second is the SI unit of angular velocity. So for a complete rotation, the angle becomes ${360^0} = 2\pi$ radian. So the angular speed becomes
$\omega = \dfrac{\theta }{t} = \dfrac{{2\pi }}{t} = 2\pi f$ ……………(4)
as $f = \dfrac{1}{t}$ , where $\theta =$ angle subtended.
So angular speed,
$\omega = 2\pi \times 2 = 4\pi$ rad/s