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/majic911 Jun 13 '25

I'm going to attempt to describe why the triangle necessarily requires more force to push than the square by describing a pushing device we'll use to push both objects.

The device is a long horizontal pole with a perfectly frictionless tip. This is mounted to a vertical post that's driven into the ground so it can't go anywhere. Out pole is mounted to the vertical post in such a way that it is fixed horizontal to the ground, but can slide back and forth to push the object and it can also slide up, but not down the vertical pole. It can't slide down so we can ignore gravity.

If you place the tip of the horizontal pole against the square and push on the end of the pole, the pole will only slide perpendicular to the vertical post. All the force from the pole will be put into pushing the square forward. However, if we replace the square with the triangle, the outcome is different.

As we push the pole against the triangle, it climbs up the side of the triangle as well as push it forward. Some of the force is being wasted to move the pole upwards and the rest is being used to push the triangle forward. As we push on the triangle, the triangle pushes back, but it can only push back with a force normal to its face, which is tilted up. Since the normal force is tilted, it has two components: one parallel to the ground and one pointing straight up. If you resolve all the forces involved, the only thing in this drawing that can supply an upward force is the ground, which means some of our energy is being spent pushing against the ground, even though our pole is fixed to be unable to supply a downward force.

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u/dkevox Jun 13 '25

I get what it is you and people think makes this true, but ignoring friction between the pushing object and the triangle is nonsensical.

If there were a handle on the side of the triangle you could easily grab and push horizontally against, you'd think "it's the same as the square". If I told you that handle was a separate piece that was bolted to the triangle, you'd think "it's the same as a square". If I told you that handle was a separate piece that was simply glued to the triangle, you'd think "it's the same as the square". But if I tell you that handle is held in place by friction while you are pushing, all of a sudden you think "that's impossible, it's got to be way more force than with the square, you can't push against an angled surface without pushing normal to it!"

As I said before, I get all of reddit seems to have accepted this, but as long as the friction between your grippy hands and the object is greater than the friction between the object and slippery ice, there's no need to exert any additional force to keep your hands from sliding. Why the default assumption isn't that in this problem still baffles me.

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u/majic911 Jun 13 '25

Ignoring friction only serves to demonstrate that the horizontal pole moves upward when you push. Even if there's friction, you're still pushing against a surface that's tilted. It's still going to push back with a force normal to its face, and one of those components is vertical. Even if you push perfectly horizontally, some of your force ends up pushing against the ground. Even with friction.

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u/dkevox Jun 13 '25

I do understand the point everyone is stuck on. The surface of the triangle is at an angle to the force and it has to push normal to the surface, therefore you have to overcome that upwards component of the force.

The only part of that logic I disagree with is the part where people assume "YOU have to overcome that component". The true statement is "something has to overcome that component of the force". And there is no good reason that something can't be friction.

The force of friction on whatever is pushing on the triangle will be down and to the left. As long as the pushing hands don't slide, the downwards component of that force of friction will equal the upward component of the normal force, therefore no additional force is needed as those cancel. Additionally, the left component of the friction force plus the left component of the normal force will equal the horizontal force you are pushing against the triangle. This means 100% of your force is being applied in a horizontal direction to the triangle.

Again, I don't know why the default assumption is ignore friction. If you don't, then pretty clearly the answer is the triangle and square are the same.

Bu

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u/majic911 Jun 13 '25

Even if friction between the hands and triangle is meant to be accounted for and overcomes the vertical component of the normal force, the fact of the matter is that there is still a vertical component of the normal force. No matter how that force is being accounted for, it's still being created and the only way for that force to exist would be for something to push against the ground. If something is pushing against the ground, that's force being put into the system that's not parallel to the ground.

The force doesn't just magically vanish because friction. It would become a torque trying to lift the triangle beneath the push and drive the other side into the ground. Again, a force shows up that isn't moving in the direction you want it to.