GrrlScientist sends a link to this rather wild stunt from India:
How is it possible? What kind of friction is necessary, and is it any more difficult for the cars to do the stunt than it is for the motorcycles?
Before we do any math, I want to think about the problem qualitatively. Let’s tally up the forces acting on the vehicles. First, there’s gravity. It points vertically downward, straight toward the center of the earth. Second, there’s the normal force. “Normal force” means the force normal (in this context a physics technical term meaning perpendicular) to the surface. It’s the force the track exerts on the vehicle. It points, as the name indicates, perpendicular to the track surface. Third, there’s friction. It points in the opposite direction of wherever the vehicle would be sliding if there were no friction.
You may be a little suspicious of that phrasing, and you should be. How do we know what direction that is? Well, we could calculate it but we don’t have to. Let’s take a look at the diagram of the situation, conveniently found on the Wikipedia article on the inclined plane:
The gravitational force downward is mg, and the normal force is N. The terms with sine and cosine are just the gravitational force expressed in components – those two forces are exactly equivalent to the downward vertical mg. We do know from Newton’s laws that F = ma, i.e,, the sum of all the forces F is equal to the mass times the acceleration. In the situation in the video, the total acceleration parallel to the track is zero. The motorcycles neither ascend nor descend once they reach their cruising altitude, so to speak. But that doesn’t mean there’s no acceleration. If there were no acceleration, the vehicles would continue in straight lines, sinking into the track like ghosts. Instead, the riders are essentially orbiting; their acceleration is directed toward the center of the circle. Their speed doesn’t change, but their direction does.
From freshman physics, we know the force required to keep an object in uniform circular motion is F = v^2/r, toward the center of the circle of radius r. But looking at the diagram, it’s easy to see that N points toward the center of the circle, and so does the parallel component of mg. The frictional force f doesn’t, and so even if there were no friction and the track were made of ice, N and mg could produce the required acceleration, keeping the riders in place.
Now I’m going to depart from the diagram just a bit to make the math easier. Instead of decomposing mg into parts with respect to the plane, I’m going to decompose N into vertical and horizontal components with respect to the ground.
Since the cars neither rise nor fall, the vertical component of N has to be equal to the downward force mg:
The only horizontal force is the horizontal component of N, and it has to be equal to the force needed for circular motion:
But we already figured out what N was, so we can substitute:
Now with we can solve for v, the speed necessary to stay orbiting the wall even in the absence of friction:
Because sin/cos = tan, and the m’s cancel out. Which is nice – it means that it’s equally easy (or hard) for the heavy cars and light motorcycles to do the trick, provided they can actually get up to that speed and provided they’re both not too big with respect to the turning radius r.
So how fast do they have to go? It’s hard to estimate what r is, but we can guess perhaps 10 meters. The angle is also tough, but let’s guess 70 degrees just as an estimate. That gives us a value of about 16.4 m/s, or 36 miles per hour. To my unpracticed eye this seems a bit high, but then I’ve probably overestimated the angle. A 60 degree incline would mean 29mph, for instance. Either way though, it’s pretty clearly doable for motorcycles and cars alike. I don’t know what the Indian equivalent of OSHA might think, but the laws of physics are fine with this bit of daring.