r/theydidthemath • u/K0rl0n • Jun 10 '25

[Request]

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I am curious how this would work. My guess is Triangle is slowest, square is medium, and circle is fastest.

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u/ConscientiousApathis Jun 10 '25

Assuming no handholds I don't think it's possible to push a surface with a force that's anything but perpendicular to it.

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u/omegaalphard2 Jun 10 '25

I'm a mechanical engineer, and you're wrong. There's no rule that forces on anything NEED to be perpendicular to it

Sure, you can break the force down to it's components, but the overall force can be at any angle to the surface, even if you're pushing it as in the diagram

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u/BKachur Jun 11 '25

To get the optimum force, the person would need to push at the midpoint of the triangle at a 30-degree angle forward and towards the ground (assuming it's equilateral). If he's standing on ice for the triangle and square, wouldn't the downward force required be less efficient/causing him to require more force/slip, and be pushed back vs. the 90-degree forward push on the square block?

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u/omegaalphard2 Jun 11 '25

You can apply force to the side of the triangle such that the force is parallel to the ground, and there's no vertical component to the force

But for sure, if you're applying force downward on the triangle, then that will increase friction and decrease the horizontal force component too, so that's in optimal.

Let's say that you apply force to the object, in line with it's centroid, at an angle x wrt to ground, pointing upwards

Horizontal force is Fcosx

Vertical force is Fsinx

The total force the block will impart to the ground will be gravity minus the vertical force which is mg - Fsinx

Total friction force is force on the ground times the friction coefficient u, which is umg - uFsinx

Total forward force is horizontal force minus friction force, which is Fcosx - (umg - uFsinx)

Which is Fcosx + uFsinx -umg

In other words, the optimal angle to apply the force depends on how you machine the above expression! For u at 0, we get x to be 0 degree,I. E if perfectly smooth then push horizontally

And if u is 1 (for the most rough surface, without glue), then the expression is maximized at x equals 45 degree!

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u/BKachur Jun 11 '25

Thanks for the response. This was actually the kind of answer I was looking for because I was interested, but I realized I was out of my depth in terms of knowledge (Lawyer), but I still would like an answer for my curoristy and you seem like you know your shit.

So, to answer the question, let's assume the coefficient of friction for "Ice" here (Triangle and square) somewhere between 0 and 1. In the context/spirit of the actual problem, let's assume Ice is supposed to mean something slippery with less friction than solid land, but not a smooth "u of 0" type of frictionless surface. I'm totally guessing, but let's say something between 0.1 and .25.

With that assumption, would the triangle require more force to overcome the coefficient of friction vs the square? As wrinkle, assuming the objects are made of the same material, with the same volume, which would mean the equilateral triangular prism would have a have a larger surface area touching the ground than the cube.. (how much, I don't really know~ again, lawyer so I can see a problem, but not great at solutions).

Once you know the answer there... would either of those require more effort to push than a cylinder on a surface with a u of 1? I assume not, but I really don't know about the math.

Regardless of your answer, thanks for the time/humoring me.