I understand that this is theoretical, and that there’s no way to create such a cable (of any significant length) in reality. Let’s consider an incline instead. Do heavier people go faster when rolling down an incline in a given vehicle?
Ah! yes. That makes sense. Otherwise we’d all be moving at relativistic speeds when going down a slight hill on rollerblades.
In the absence of aerodynamic drag, no, Fat Guy and Skinny Guy will accelerate at the same rate, just as they would in freefall. That formula I cited earlier, a = 9.81*sin(theta)? Notice that the mass of the object is not in that equation.
But with aerodynamic drag, the Fat Guy will travel faster than the Skinny Guy. Probably just pointing out the obvious, but that’s my job.
As you note, this is true in the absence of air. Otherwise the final speed that results from an inclined path will be lower, due to a longer path that requires pushing more air molecules out of the way (and the small but non-zero contribution of pulley friction).
Chronos, you mentioned actually testing the hill hypothesis in sledding. Do you also attribute the difference to wind resistance?
And, for curiosities sake, how big are the differences?
Quite right. I should have noted that your formula is the more general form, covering paths at any angle.
Yes, I attribute the difference to wind resistance, since I can’t come up with anything else that could account for it: All other forces on the sleds would be directly proportional to the masses, and therefore produce a mass-independent acceleration.
And as for how big the differences were, there wasn’t any speed-measuring equipment available, nor were the distances along the sled tracks marked, so I don’t have any numbers (and I would probably have forgotten them by now even if I had them). So I can’t say anything more than that it was qualitatively extremely significant.
Was the surface upon which they were sliding snow? I was at some point led to believe (by a ski instructor) that a snow surface actually changes composition (i.e. packs down and melts a bit) when you put weight on it, and therefore doesn’t obey the usual “friction is proportional to normal force” rule.
It was a toboggan track, well-used, so any packing that was going to happen had already happened long before either test sled ever touched the track. Though I’ll grant that “friction proportional to normal force” is only ever an approximation for any surface, and it might well be a poor approximation for ice-- I don’t know the practical details on that one.
A piece of news that may be relevant to this thread: Tourist breaks back on Sentosa ride Tourist breaks back on Sentosa ride At least he can move his legs now, I thought a broken back is guaranteed paralysis.
I did the math and it appears that for fairly typical pulleys, there’s somewhere around 15 joules of energy in the rotating masses @ 50 mph.
Assumptions:
50 mph speed, or 22.35 m/s
pulley diameter 21 mm (from here)
2 pulleys of 60 g each (estimated based on 270 g total trolley mass and materials)
Energy in a rotating disk is ¼Mr²ω²
I get 14.46 J. So yeah, not significant here.
No guarantee. It’s possible to break your spine without damaging your spinal cord; the latter is what causes paralysis. Twelve years ago my brother was in a motorcycle accident in which he broke his back; there was no spinal cord injury though, so he never experienced any paralysis.