Angular quantities, in its bsic form, are defined in very geenral 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 or interpreted.

It must be understood that we can define and interpret angular quantities very generally with respect to a point in the reference system. For some, it may sound a bit inconsistent to know that we can actually associate all angular quantities even with a straight line motion! For example, we can calculate torque on a particle, which is moving along a straight line. We shall work out with appropriate examples to illustrate the point.

Indeed, angular quantities are found to be suited to 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, but as a student of physics it is important to know the complete picture.

There are many such situations in real time, which can not be classified as a particular motion type. Consider projectile motion along a parabolic path. We employed the concept of independence of motions in mutually perpendicualr directions, which is an experimental fact. This paradigm of analysis suited the problem in best fit manner.

However, we can equivalently think of this motion even in terms of angular quantities like a torque about certain axis, which keeps shifting. If we observe closely a projectile path, then we can see that the parabolic path of a projectile is actually made up of many circular arcs along which the projectile undergoes rotation for brief periods.

We must understand here that the broadening the concept of angular quantities is not without purpose. We shall find that the de-linking of angular concepts like torque and angular momentum from a point (instead of 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). Importantly, we can employ angular quantities to analyze situations, which are not strictly rotational (like motion along a straight line).

### 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.