Undoubtedly this has been posed before in GQ, but humor me, if you would, with a step-through of the basic dynamics of the loop (this kind of thing).
I was thinking of aircraft banking envelopes, and at the same time imagining how on the throttle (either vehicle) the engine output could be feathered down as the ballistic forces factored in and out. And then rocket science (oy…).
In order of simplicity, given ideal energies (the high school physics lesson I’d like), how do you figure out what will satisfy the loop requirements so that sufficient energy is applied normal to the any track to keep positive (over G) force?
Given a real motorcycle, what is the maximum radius?
Or, different takes: given a radius, minimum initial and average velocity?
And given a radius, if you wanted the most gas efficient trip, what would be the speed/mph(?)/rpm(?) during the course of the loop-de-loop?
And–here I’m totally clueless, and its ancillary to specific OP–What would be a street-conceivably-legal motorcycle engineer’s options say he was given unlimited money to design for one doable big loop?
Rocket scientists and aircraft engineers do this kind of thing for breakfast, in undergraduate school, right?
The math is easy for a constant-radius loop: You set the centripetal acceleration equal to g at the top, and that gives you the speed you need at the top. Then you use conservation of energy to find the speed needed at the bottom, to give that speed at the top.
The catch is that most real loops aren’t constant radius. It’s generally easier on the rider to have a loop that’s teardrop-shaped, which also incidentally decreases the needed speed. Just how pinched you want the top, and precisely what shape you use for the rest, will depend on a lot of complicated engineering factors, like the length of the vehicle’s wheelbase, the height of the vehicle and/or rider, and the rider’s g tolerance.
As Chonos says, if you’re dealing with a purely passive frictionless vehicle coasting, you just need enough speed at the top for centrifugal force to exactly offset gravity. Get it moving fast enough at the bottom so it slows to that target speed at the top and let it loose and it’ll be fine going around and around forever. In the real world with friction it’s not quite that simple.
Here’s another facet to consider:
In the entire bottom hemisphere you don’t actually need any speed at all. Gravity is pulling you towards the sphere’s wall surface and needs no boost from centrifugal force. So a vehicle could drive around the bottom half at 1 furlong per fortnight (ff[sup]-1[/sup]) even though it needed to be going, say, 50mph at the top to stick to the sphere there. Assuming it was coasting rather than powering all the way around it would need to be magically instantaneously boosted from 1ff[sup]-1[/sup] to, say, 75mph as it passed vertical going uphill so as to still be going 50 over the top then passively accelerate back to, say, 75 at the vertical going down. At which point it can be magically de-boosted back to 1 ff[sup]-1[/sup] for its next traverse of the bottom half. Lather rinse repeat.
And a third:
For a real motorcycle in a real loop we need to add in its power, top speed, and traction information. Traction matters because actual traction is proportional to normal force = gravity +/- the centrifugal force. If the net “weight” of the motorcycle against the sphere is only a few ounces when going over the top, the tires will slide if anything other than a tiny amount of power or steering effort is applied.
There’s a pretty complicated pile of differential equations to solve to actually work that out with real numbers for a real motorcycle.
If you make the sphere small enough and make the bike powerful enough it works with plenty of margin. The hard part comes in answering Leo’s question about what’s the limit case and how do you know it’s the limit case.
The reason so many ancient buildings still stand is those are the ones where the “engineers” of the day left in lots of structural margin. Not by design, but by accident. Shame about what that did to the patron’s budget.
The ones that aren’t still standing are the ones where the “engineers” didn’t leave in enough margin. Not by design, but by accident.
The difference today is that we know how to calculate those margins and how to build to our calcs. Which is the most economical way to do it.
That doesn’t mean backyard engineering doesn’t still work given enough margin and/or enough blood. As any number of redneck “reality” TV shows show us.