Let us think that the wheel of a car which is pure rolling on the road is of square shape, not circular. It is a bit wierd but lets still think so. We consider the motion of that square from the instant the side AB touches the road-surface fully, to the instant side BC touches the road-surface fully, as shown in the figure. During this time interval, the contact-point B remains fixed on the ground and it is as if the whole wheel rotates about the axis passing through the contact-point during this short segment of the wheel's journey. It is amazing to find that even the c.o.m. is describing a circular path during this time. After this time-interval, the contact point changes and so the axis also change. Again, during the next time interval, the whole wheel rotates about this new axis. So, as seen from the road frame, this translating axis passing through the contact point is the perfect axis for the rotation of the wheel. Instead of a square, by taking any polygon (regular or irregular). we will find similar results. But as the number of sides increase, (area of the wheel remaining constant), the lifetime of each axis decreases. Now, we know that the circular wheel is an infinite sided regular polygon. So, the contact point of the circular wheel remains fixed on the road for an infinitesimal time interval, and during this time interval, the whole wheel (even the c.o.m) rotates about the axis passing through this contact point.
But a wheel is a wheel and if it is rotating, it is supposed to rotate about its axle. As seen from a frame of reference fixed w.r.t. the c.o.m. of the wheel, the wheel surely rotates about its axle. That is if we see only the wheel and we travel with the same translational velocity as the wheel, then the axle will appear to us as the axis. But if we stand on the road and witness the motion of the wheel on the road, then the axis passing through the contact point appears to be the more perfect axis of rotation. In the first case we consider the fact that the wheel is fixed about the axle, and in the second case we consider the fact that the wheel is fixed at its contact point on the road for an infinitesimal time. So, both the axis are natural axis and the angular variables about the two axis will be different from each other.
In case of pure rolling the equation K.E. = 1/2 I 
2 can be directly used to find the kinetic energy of the wheel if I is the moment of inertia of the wheel w.r.t. the axis through the point of contact.
Acknowledgements: I got the basic idea behind this from sss (see the 6th post in http://www.goiit.com/posts/list/15/4000.htm ), and also took help from H.C. Verma's 'Concepts of Physics'. So, thanks to both of them.
Back to Community Shelf
15 Nickels awarded!![[Post New]](/templates/default/images/icon_minipost_new.gif){kind=icon}
![[Thumb - sqwhee copy.gif]](/upload/2007/3/3/87621f561da73d1ee0effc412071a7b2_4587.gif_thumb)
21
