We all know that angular displacement represents the rotational effects of a body and thus we can say that it's an axial vector direction of which can be determined using right hand principle. However, the trouble comes when we try to perform vector addition on two angular displacements. Refer to the figure given below:
The given object (a book) has been rotated 90 degrees about horizontal and vertical axes. It's clear that final configuration is different depending upon whether it was rotated about horizontal axis or vertical axis in the first step. Thus addition of the two angular displacements is not commutative i.e. `A+B!=B+A`. However, two vectors always follow commutative law of addition. Thus in this case we conclude that angular momentum is not a vector quantity.
Now consider a similar situation but instead of 90 degrees, the book is rotated only 45 degrees in a similar fashion as above:
Here you can see that although the addition is not commutative, but still they are coming closer to it. Now, consider the third case with 10 degree rotation:
In this case, you can't tell the difference with your naked eyes whether the addition is commutative or not. The final position of both books is almost exactly same in both the case. This can be interpreted as follows:
No doubt, angular displacements are not vector quantities when they are large or strictly speaking, finite. However as they approach infinitesimally small values, they do show vector-behavior and follow commutative law of vector addition. Thus we can conclude that angular displacement is a vector if infinitesimally small and not a vector if finitely large.
No comments: