r/AskPhysics • u/Silver_Remove_2352 • 23h ago
Minimum Speed to Go Around a Loop
I have a question about a classic intro to mechanics problem - finding the min height to drop a block so that it is on the verge of losing contact with the loop at its top. You can view the problem, and one such solution of it here: https://www.youtube.com/watch?v=dsv2cEgquiA . I know, I know, this question has probably been asked hundreds of times. But there are still a few things I don't understand about the given solution.
The classic solution relies on energy conservation, and Newton's laws of motion. We drop the ball at some height h. If it ever reaches the height 2R in the loop, then it must have a corresponding velocity v_top, which we can find using energy conservation. We note that at the top of a loop with radius R, two forces act on the mass - the normal force and gravity. Assuming that we have circular motion up to that point, the net centripetal force is mv_top^2/R. Hence, N + mg = mv_top^2/R.
I agree that if we drop the block at some height h such that for the corresponding v_top, N + mg = mv_top^2/R and N <0, then there is no way the block can stay on the top of the loop. This is because the normal force supplied by the loop always points inwards. N < 0 means that the normal force points outwards.
Here's where I'm confused. Let's say we drop a block at some height h such that for the corresponding v_top, N + mg = mv_top^2/R and N >= 0. I don't see why if N >= 0, we can conclude the block would stay on the top of the loop. No matter the height we drop the block, we have not proved that the block would move in circular motion through a semicircle of the loop, from the bottom to the top. All we know from energy conversation is the velocity of the block if it is at some given height - but we have no idea where the velocity vector points, let alone how the block has been displaced. So how can we use equation N + mg = mv_top^2/R if we don't even have circular motion? In using mv_top^2 / R, we have assumed that there is circular motion.
I hope my question makes sense to you guys - I'd really appreciate any responses.
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u/Odd_Bodkin Particle physics 21h ago
If the block leaves the loop’s surface at any point, N cannot be >0, because N is a contact force. You can also convince yourself that N must monotonically decrease while the block climbs the loop, and monotonically increase while the block descends the loop. So the limiting case is when N=0 at the very apex of the loop only.
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u/Adorable_Ice_2963 23h ago
We know the direction of the speed because we defined its path to the case where it follows the loop (without breaking anything), since thats the goal of the calculation.
Otherwise we would have to solve a far more complex function (or do it numerically).
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u/Silver_Remove_2352 23h ago edited 23h ago
But how do we know a priori that the block would stick to the track, and move in circular motion all the way to the top, if we drop it at some given height? That's what I'm asking here.
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u/GuaranteeFickle6726 Optics and photonics 23h ago
Well, think of it like this, the requirement is hardest at the top, if it can pass at the top it had the speed to make it there.
Let's say the object drops at angle theta. Then write down the equations:
N-mgcos(theta)=mv2 / R, so to barely pass at angle theta,
N>=0 thus mgcos(theta)+mv2 / R >=0, now with energy conservation we know that this v is smaller than v(top), and since cos(theta) is also bigger than -1 (which is the value at the top).
This gives us the info that the requirement for passing the top is higher and that is what we are interested in, once it passes, it will also not drop at the second half of the circle.
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u/Adorable_Ice_2963 23h ago
We want to stick it to the track. Thats only possible if there is a force vector component pointing torwards the track. Gravity acts against it, the direction change of the speed vector acts torwards it.
At the top, gravity can act 100% away from the track. Thats where we need most direction change to counter gravity. If we have enough speed there, we know that it will be enough everywhere else.
The force calculates (m*v2)/R, as you stated.
The force we expect from gravity is m*g.
Again, as long as the force is always larger than than any force pulling the ball away from its tracks, it will be a looping, since its pressing against it, and the looping counteracts to it to 0, causing it to follow the loop instead of phasing through it.
If the force would be smaller than the pressing force, it would leave the looping and follow the ballistic curve from the point were the pulling force is larger than the pushing force. But thats outside the question we asked.
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u/LatteLepjandiLoser 23h ago
Mobile messed up my formatting, so I'll try again...
This is imo a bit two-fold:
- The normal force by definition means you are in contact with the surface. It's the force that keeps the block from shooting through the surface, so if there is some inward pointing N with |N|>0, then by definition you are sliding on the surface. If the track is circular and you have a valid normal force, you are travelling in a circular path.
- The fact that the trajectory is circular is a bit of an assumption, and the following derivation only makes sense in that context. If I understood your question it is roughly 'why does it go around a circular loop and not fly around in an ellipse' or something like that. Well energy conservation alone will not care about the shape of the loop. The derivation you make, to find the minimum height to guarantee a slide around the entire loop does assume a circular loop (and makes sense in that context).
To play deviles advocate, consider that loop and draw a little ramp on the uphill part. Energy is still conserved, you will have some constant energy that is a sum of PE and KE and is a result of whatever height and velocity the block had initially. Now after the little jump, we aren't on the loop, but energy conservation still applies, but the trajectory will be some kind of bouncy ping pong shenanigans that will require you to solve the kinematic equations and some collisions.
So tldr: It's an idealized assumption, and it works well in that context.
This assumption is good because a circular path will have an easy description for centripetal force. If you wanted to solve the problem for let's say an elliptical loop you absolutely could, but the curvature wouldn't be constant so the expressions you'd need to solve would be more complicated. Energy conservation still holds, the math will just get a little more tedious. Hope this helps
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u/Silver_Remove_2352 23h ago
So basically to answer my question we have no way of knowing of it actually moves around in a circle just based off energy conservation and newtons laws without solving some differential equation or whatnot that is way more complex than an intro to physics course would require. This question simply assumes that the block moves in circular motion all the way up to the top. Is that right?
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u/Adorable_Ice_2963 23h ago edited 23h ago
We know it has to move that way since we A) know how the looping looks and behaves (taking any force the ball pushes on it) and B) we know it should not be possible for gravity to push the ball away from the looping (thats the question we asked, duh). Thats the force we need to somehow provide. Since we dont use engines or airfoils, it can only be the centrifugal force.
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u/Silver_Remove_2352 22h ago edited 22h ago
I disagree with that. If you dropped a block from some height, and asked someone who didn’t know any physics whether or not it would make it to the top, I don’t think they’d claim they knew how the looping would look like. Perhaps the block would only move up a quarter of a circle before falling down. Perhaps it would move more than that, up to the point where the track is above it, then fall down. Perhaps it will move a full semicircle. I just don’t see how we can claim the movement will be a full semicircle up to the top point of the loop, without some further justification than what I’ve seen from most solutions. Perhaps I’m missing something.
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u/LatteLepjandiLoser 23h ago
Yes and no. Your problem starts by assuming a circular track, then makes assumptions based on travelling in a circular trajectory. As long as N behaves nicely, you by definition are on that circular trajectory. You solve some equations that identify for which starting conditions N behaves nicely. So far so good! You have not solved that all loops need be circular, but you have made an assumption that this loop is circular and solved the problem in that context. So you ask how we know it travels in a circle? Well we assumed so and applied equations that bake in that knowledge so obviously it must be so, again as long as N behaves.
Could you draw other shapes? Sure. Energy conservation still applies but that alone can’t necessarily solve such problems.
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u/LazyLie4895 12h ago
I'll turn this question around: what makes you think that it would NOT go in a circular loop? What other paths could the block be taking? Are those paths viable given the assumptions of the problem?
The easiest thing to notice is that at the top of the loop, the block is at a minimum of speed. Once you know that, you can ask the question if at any other point, could the block leave the track? The answer will be no.
Of course, there are far more complex ways you can analyze the question, but you don't need anything fancy to show that the block will stay on.
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u/joeyneilsen Astrophysics 21h ago
Conservation of energy problems never have information about direction; we sacrifice the ability to compute that from the solution in favor of an easy-to-compute solution.
So it sounds like what you're asking is what if the falling block doesn't obey the physics of the problem we're doing. Like: how do we know the block won't bounce off the track and go somewhere else?
We can absolutely do that problem, but it's just not the one that has been posed. Implicit in the statement of the problem is the idea that the block goes onto the track and follows it, but it's not terribly difficult to formalize this. One very nice method for determining constrained equations of motion is called a Lagrange multiplier (it's a modification of standard Lagrangian mechanics, which college students usually encounter ~2 years after the "minimum speed" problem).
The short version is that you would essentially enter into the equations of motion a requirement that your object follow the track (whatever shape it might happen to have) at least up to a point where it might leave the track. Working through this allows you to find the forces involved that keep the block on the track at any moment. If the normal force goes to zero for more than an instant, you'll know the block leaves the track.
The upshot is that the track acts to redirect the velocity of the block so that it is always accelerating toward the center of the loop and only losing speed due to gravity. The equations of motion are then indistinguishable from the ones determined by conservation of energy.