Question

The shaft of a motor rotates at a constant angular velocity of $3000rpm$. The radians it has turned in $1\sec$are.
(A) $1000\pi$
(B) $100\pi$
(C) $\pi$
(D) $10\pi$

Answer 1

In order to solve the problem we need to know about angular frequency, its formula , its unit , and a little bit of knowledge about shafts.
Shaft: A shaft is a rotating machine element, usually circular in cross section, which is used to transmit power from one part to another, or from a machine which produces power to a machine which absorbs power.
Angular frequency: It refers to the angular displacement per unit time
$\omega = 2\pi f$
Where $f$ is the frequency
As we know
$f = \dfrac{1}{T}$
Where $T$ is the time period

Complete step by step solution:
Given $\omega = 3000rpm$
$rpm =$ rotation per minute
To convert it into rotation per second$(rps)$
$\omega = \dfrac{{3000}}{{60}}rps$
$\omega = 50rps$
To convert it into radian per second
$\omega = 50 \times 2\pi \,rad/\sec$
$\omega = 100\pi \,\,\,rad/\sec$
Then we can say that the radian it turns in 1 sec is $100\pi$

So the correct option is (b).

Note:
Rotational speed is not to be confused with tangential speed, despite some relation between the two concepts. Imagine a rotating merry-go-round. No matter how close or far you stand from the axis of rotation, your rotational speed will remain constant. However, your tangential speed does not remain constant. If you stand two meters from the axis of rotation, your tangential speed will be double the amount if you were standing only one meter from the axis of rotation.
Revolutions per minute is the number of turns in one minute. It is a unit of rotational speed or the frequency of rotation around a fixed axis.