Question

The angular speed of a flywheel making 180 r.p.m. is

$${\text{A}}{\text{. 2}}\pi {\text{ rad/s}} \\\ {\text{B}}{\text{. 4}}\pi {\text{ rad/s}} \\\ {\text{C}}{\text{. 6}}\pi {\text{ rad/s}} \\\ {\text{D}}{\text{. 3}}{\pi ^2}{\text{ rad/s}} \\\$$

Answer 1

The angular speed of a rotating object is given as the product of $2\pi$ with the frequency of the rotational motion of the object. We are given the frequency of the rotation of the flywheel and using this formula we can calculate the angular speed of the flywheel.
Formula used:
The angular velocity is given in terms of frequency of the angular motion by the following expression.
$\omega = 2\pi \nu$

Complete answer:
The angular frequency of the rotational motion of the flywheel is given as
$\nu = 180rpm$
First we need to convert the units from rotations per minute into rotations per second. It can be done by using the information that in one minute we have 60 seconds. Doing so, we get
$\nu = \dfrac{{180}}{{60}}rps = 3rps$
Now we can calculate the angular speed using the following formula.
$\omega = 2\pi \nu$
Inserting the value of frequency, we get
$\omega = 2\pi \times 3 = 6\pi {\text{ rad/s}}$
This is the required value of the angular velocity and hence, the correct answer is option C.
Additional information:
The angular velocity is the rotational counterpart of the linear velocity. The relation between the angular velocity and the translational velocity is given as
$v = r\omega$
Here r signifies the radial distance of the rotating body from the axis of the rotation of the body.
Similarly, we have rotational counterparts of other translational quantities as well. For example, the rotational counterpart of mass or inertia is the moment of inertia, the rotational counterpart of linear force is the torque and so on.

Note:
It should be noted that the units of frequency should be in rotations per second. Also, while changing the units we divide the frequency by 60 because the units mean rotation ‘per’ second which means that the time is coming in the denominator.