Angular quantities, in their basic forms, are defined in very general context. When we defined, for example, torque in the context of rotation of a rigid body or a single particle attached to a "mass-less" rod, we actually presented a definition in special context of a fixed axis. As a matter of fact, the definition of angular quantities does not require an axis to be defined.

We can define and interpret angular quantities very generally with respect to a "point" in the reference system - rather than an axis. This change in reference of measurement allows us to extend measurement of angular quantities beyond angular motion. For some, it may sound a bit inconsistent to know that we can actually associate all angular quantities even with a straight line motion or a translational motion! For example, we can calculate torque on a particle, which is moving along a straight line or we can determine angular displacement and velocity for a projectile motion! We shall work out with appropriate examples to illustrate the point.

Indeed, angular quantities are found to suit rotational motion or where curvature of path is involved. For this reason, we tend to think that angular quantities are applicable only to rotation. There is nothing wrong to think so. As a matter of fact, there are many situations in real time, which suits a particular analysis technique. Consider projectile motion along a parabolic path. We employ the concept of independence of motions in mutually perpendicular directions, based on experimental facts. This paradigm of analysis suites the analysis of projectile motion in a best fit manner. But as a student of physics, it is important to know the complete picture.

We must understand here that the broadening the concept of angular quantities is not without purpose. We shall find out in the subsequent modules that the de-linking of angular concepts like torque and angular momentum from an axis, lets us derive very powerful law known as conservation of angular momentum, which is universally valid unlike Newton's law (for translational or rotational motion).

The example given below calculates average angular velocity of a projectile to highlight the generality of angular quantity.

### Example 1

Problem : A particle is projected with velocity "v" at an angle of "θ" with the horizontal. Find the average angular speed of the particle between point of projection and point of impact.

Solution : The average angular speed is given by :

Angular velocity |
---|

Here,

From the figure, magnitude of the total angular displacement is :

Putting these values, we have :

From this example, we see that we can indeed associate angular quantity like angular speed with motion like that of projectile, which is not strictly rotational.