# How can I calculate rotational speed?

Aug 4, 2018

Hope it helps...

#### Explanation:

Rotational speed $\omega$ can be evaluated considering the change in angular displacement $\theta$ with time $t$.

$\omega = \frac{\theta}{t}$

Practically this can be a bit dfficult so what we do to measure the rotational speed of, say, a motor shaft is to visually count the number of rotations per minute.

We observe a point on the rim of the rotating shaft and we count the number of rotations (say 10) and check the time elapsed.
We divide the number of rotations by the time and get the rotational speed in, say:

$\text{revolutions"/"minute}$

Now, we can improve a bit by considering that during a complete revolution the angular displacement will be $2 \pi$ radians so we can convert it into something we can use in mathematical evaluations:

$1 \text{rev"/"min} = \frac{2 \pi}{60} \frac{r a d}{s} = 0.105 \frac{r a d}{s}$

So now we can go from revolutions per minute to angular dsplacement $\theta$ per second.

Consider an example:
the shaft of a motor rotates $1800 \text{rev"/"min}$. Through how many radians does it turn in 18 s?
The rotational speed in radians per second that corresponds to $1800 \text{rev"/"min}$ is:

omega=(1800"rev"/"min")(0.105((rad)/s)/("rev"/"min"))=189(rad)/s

since $\omega = \frac{\theta}{t}$, in 18s the shaft turns through:

$\theta = \omega t = 189 \cdot 18 = 3402 r a d$

[Aside: remember that once we have the angular displacement in radians it is only a matter of considering the radius $r$ to get the speed of each radial point]