November 3rd, 2014, 12:14 AM  #1 
Senior Member Joined: Apr 2014 From: UK Posts: 892 Thanks: 328  Rolling wheel hits a gap
I'm trying to work out the deviation in acceleration of a wheel travelling along a perfectly horizontal beam and goes over a gap, but I'm struggling to get started with it. The momentum is such that the wheel doesn't slow down in its horizontal motion, but the rotational speed will increase as it drops into the gap and then decrease as it rises out of the gap. The gap is such that the wheel can ride over it quite easily, the radius of the wheel is much larger than the gap. Do I start by working out the equation for the vertical position of the wheel with respect to the beam? 
November 3rd, 2014, 05:55 AM  #2 
Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116 
How well have you thought this through? It seems to me that the most significant factor is the speed of the wheel. If the wheel is moving quickly, the gap will have little effect (as anyone who rides a bike will know). Also, I don't see how the angular velocity will increase (unless the wheel is moving very slowly) as it drops into the gap. Think about a wheel going over a cliff. The main change to motion will be a impact on the far side of the gap. The other significant factor here is the coefficient of restitution. If the gap is large enough, then the wheel will bounce up off the beam (this, again, is what would happen on a bike). But, if the gap is small, there may be very little change in motion. 
November 4th, 2014, 12:27 AM  #3 
Senior Member Joined: Apr 2014 From: UK Posts: 892 Thanks: 328 
I think you are right, the gap will have little effect, some experimentation with excel shows a 1% increase in speed of rotation for a 0.5m radius wheel and a 0.14m gap. This was deliberately an oversized gap, just to get some bigger results (assuming my excel equations are correct, I've used small time increments to get close to the answer) The angular velocity will increase as the wheel rolls into the gap as it has further to travel, albeit not very much. The wheel isn't dropping down into the gap, which is why the cliff analogy doesn't apply in this case. Obviously there will be a speed at which the wheel cannot roll into the gap, which might make in interesting 2nd part to the problem Also, for the purposes of the problem, the beam is high off the floor making the gap deep enough that the wheel never touches it, it only has contact with the beam. 
November 4th, 2014, 12:47 AM  #4 
Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116  If there were no gravity, the wheel would move in a straight line over the gap. Gravity is precisely what causes the wheel to drop (slightly) into the gap and hit the edge on the far side of the gap.

November 4th, 2014, 03:09 AM  #5 
Senior Member Joined: Apr 2014 From: UK Posts: 892 Thanks: 328 
The previous sentence states that the wheel is rolling into the gap (due to gravity), it's not going over a cliff and dropping.

November 4th, 2014, 04:41 AM  #6  
Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116  Quote:
If the wheel is rigid, it won't. And, it's a much harder problem altogether to model the motion of a deformed wheel under load across a gap.  
November 4th, 2014, 05:43 AM  #7 
Senior Member Joined: Apr 2014 From: UK Posts: 892 Thanks: 328 
It was stated that the horizontal speed remains the same, so for it to roll downwards on the first corner of the gap, it must turn faster as the surface of the wheel has to travel further than rolling horizontally.

November 4th, 2014, 06:21 AM  #8  
Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116  Quote:
If you go back to the cliff problem. It is the same situation falling into a gap or off a cliff, until the wheel hits the far side of the gap. A rigid wheel will simply fall into the gap without rotating any faster. An object such as a fourwheeled vehicle would tip into the gap, because the centre of gravity is still short of the gap when the front of the vehicle starts to fall (causing an angular acceleration). But, the wheel starting to fall coincides with its centre of gravity moving over the gap. So there will be no angular acceleration.  
November 4th, 2014, 08:16 AM  #9 
Senior Member Joined: Apr 2014 From: UK Posts: 892 Thanks: 328 
Ah, I think I know where the problem/confusion with my explanation lies, I've left off the rest of the vehicle.... A very heavy bike with rigid wheels will do, or a steamroller (neither with steering ) Other assumptions are that it's rear wheel drive of constant speed and infinite grip. 
November 4th, 2014, 08:36 AM  #10  
Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116  Quote:
 

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