Sagitar Posted August 17, 2012 Report Share Posted August 17, 2012 I took this picture last week, as part of a set concerned with cycling. A person who was looking at the set asked how it was possible for the wheel spokes to be clear at the bottom of the wheel but blurred at the top of the wheel - surely he said, the wheel must be rotating at constant speed over the whole of its diameter. I tried to explain that on a moving vehicle the periphery of the wheel does not describe a circular locus about the wheel axle and we got into an interesting discussion from which it was clear that though he has been driving for many years, he hadn't any idea what happened to wheels in motion. The most difficult bit to get across, was the idea that instantaneously, at the point where the wheel and the road are in contact, the periphery of the wheel is stationary. It follows from that at the point on the periphery of the wheel that is furthest from the road surface the speed of the periphery is twice that of the vehicle. Once I got him to understand that, it was quite obvious why the spokes in the picture are clear at the bottom and blurred at the top. Quote Link to comment Share on other sites More sharing options...
andyelcomb Posted August 17, 2012 Report Share Posted August 17, 2012 Woah! That's heavy! I'm not sure I still understand that bit about "on a moving vehicle the periphery of the wheel does not describe a circular locus about the wheel axle". Is it possible to explain that to me in words with fewer syllables? Quote Link to comment Share on other sites More sharing options...
Sagitar Posted August 17, 2012 Author Report Share Posted August 17, 2012 In geometry, a locus is a set of all the points that share a property. If a wheel is jacked clear of the ground and spun, a point on the periphery will describe a circle, which is the locus of all the positions that the point on the periphery will occupy in a revolution of the wheel. If however the axle is moving relative to the ground, the locus of a point on the periphery will be a curve that combines the rotation of the wheel and the movement of the axle. The mathematics is complex, but it is easy to plot such a curve physically if you divide the movement of the wheel into angular segments and remember that the point on the periphery is always the same distance from the axis. If you start with the peripheral point on the ground, it will rise vertically at first and then swing over into a long curve passing through a half-way point at a height equal to the diameter of the wheel and return to the ground down a curve that is a mirror image of the rising curve, finishing at a distance from the starting point equal to the circumference of the wheel. This locus is clearly not a circle, but is know as a cycloid curve. If you then consider the horizontal velocity of the peripheral point as it traces the cycloid locus, its average velocity is equal to the velocity at which the axle is moving, but its instantaneous velocity varies between zero, when it is touching the ground, and twice the axle velocity as it passes through the half-way point. Perhaps to bring it back to cars, I should have said that if you are doing 70 down the motorway, the tops of your wheels are doing 140 . . . . Quote Link to comment Share on other sites More sharing options...
andyelcomb Posted August 17, 2012 Report Share Posted August 17, 2012 Ooh, so it does! Much easier to understand with diagrams, but my brain still hurts. That's why I never got any further than simultaneous equations at school - and even then could never properly understand when you would use them. I have a real problem visualising such exercises so always need someting tangible to help me understand. Thanks for the education - I'll bet you've got lots more like that up your sleeve Sagitar. Quote Link to comment Share on other sites More sharing options...
Rich Posted August 17, 2012 Report Share Posted August 17, 2012 Interesting thread. Not sure I fully understand it, I'll read it a few more times. Quote Link to comment Share on other sites More sharing options...
Tony Posted August 18, 2012 Report Share Posted August 18, 2012 Wow...... Quote Link to comment Share on other sites More sharing options...
hms Posted August 21, 2012 Report Share Posted August 21, 2012 I'm too thick to understand, but: "but its instantaneous velocity varies between zero, when it is touching the ground, and twice the axle velocity as it passes through the half-way point." You cannot have one part of the wheel velocity at zero as it touches the ground and another part at twice the axle velocity as it passes through the half way point. If one is zero then the other should be zero! So what you are saying is that the lower portion of the wheel rotates to the rear then has the forward momentum of the axle subtracted from it. The top of the wheel rotates forward and then has the forward momentum of the axle added to it. Does that make sense? I am thinking off the similarity of the doppler effect heard in police sirens, where the pitch changes as the car approaches and the goes away? h Quote Link to comment Share on other sites More sharing options...
Sagitar Posted August 21, 2012 Author Report Share Posted August 21, 2012 Not at all. The road is stationary. The contact patch on the tyre is in contact with the road and (in normal driving conditions) is not sliding on the road and must therefore also be stationary. The axle is moving at the speed of the vehicle. The wheel is pivoting about the contact point on the road. The top of the wheel is twice as far as the axle from the contact point and its horizontal speed must therefore be twice the speed of the axle and the vehicle. Andyelcomb's graphic illustrates it well. Try (or visualise) it with a stick. Mark a point on the stick to represent the axle. Mark another at twice the distance to represent the top of the wheel. Hold the stick vertically with the end in contact with the ground. Move the axle point forward slightly and see what happens to the upper point. The problem that we have with this concept is that it is not something that we can easily see. Cognitive psychologist have long been using it as an example and philosophers were debating it long before the motor car. We see a car in motion and it is "obvious" that the wheels are rotating about the axles and that the periphery of the wheel is rotating at uniform velocity. That is true if you only consider angular velocity. But forward motion of the vehicle utilises only the instantaneous horizontal component of angular velocity and that component varies from zero, at the ground, where all the instantaneous motion is vertical, to twice the vehicle speed at the top of the wheel where all the instantaneous motion is horizontal. Quote Link to comment Share on other sites More sharing options...
andyelcomb Posted August 21, 2012 Report Share Posted August 21, 2012 Would visualising a tank track help? I know its not quite what is being discussed and its rather an exaggerated example, but looking at that in motion makes it easy to see that the lower part of the track is stationary on the ground, but the top half is whizzing along. If you laid your hand in a sandy dip in the ground and let the tank track pass over it with the tank doing 30mph all that would happen is that your hand would get squashed into the ground and (maybe) not be particularly harmed. But if you were to grap hold of the upper part of the track as it passed you at 30 mph it would probably rip your arm off! Do you think I'm on the right track Sagitar? (Pun intended) Or am I in for a fascinating lecture on a completely different aspect of motion? Quote Link to comment Share on other sites More sharing options...
Sagitar Posted August 21, 2012 Author Report Share Posted August 21, 2012 I think you are on the right track . . . . It's a good analogy and easier to visualise than a car wheel. Nice one; I wish I had thought of it. Quote Link to comment Share on other sites More sharing options...
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