Physics Puzzle: Famous Mechanics “Paradoxes”
June 21st, 2009
| Categories: Mechanics, Physics Puzzles
Level of Difficulty: Highschool
I probably won’t get around to updating this blog for the next couple of days, so I’m leaving you with four puzzles in one. Here are four “paradoxes” portraying seemingly impossible scenarios. Can you explain them or reason what will happen?
- A common argument made by freshmen goes like this: to walk your feet must exert a force F (due to friction) on the surface you’re on; but according to Newton’s third law, the surface should exert a force equal in magnitude and opposite in direction. The two forces should sum to zero, so how is it possible for you to move?
- A lazy sailor decides to propel his boat by tying a sail and using a fan to blow on it. Practical considerations aside, is this idea sheer nonsense or will his ship actually move forward?
- A guy is standing on a platform. The platform is connected to a rope which goes through a pulley. The guy is holding the other end of the rope in his hand. The platform isn’t attached to anything except the rope. Will the guy be able to pull himself up by pulling on the rope, or will Newton’s third law prevent this from happening? Assume the rope and pulley are massless. Neglect friction.
- The guy from question 3 is back for more. This time the rope is wrapped around a pulley above the guy’s head and its other end is tied to a brick which weighs the same as the guy. What will happen to the brick when the guy starts climbing up the rope? As before, neglect friction and assume the rope and pulley are massless.
Enjoy!
ADDED, 24/June/2009: a solution has been posted here.
I can get the first two, at least.
1. You can’t add these two forces, because they are acting on different objects. The normal force exerted on you by the surface keeps you from falling through the floor and allows you to walk.
2. It won’t work. The fan exerts a force forward on the air, which (by Newton’s 3rd law) exerts a force backward on the fan, which is attached to the boat. If the sail weren’t there, the boat would move backward. When the sail is added to catch the air, the air exerts a forward force on the sail (which is also attached to the boat), which, under ideal circumstances, would be identical to the force exerted on the air by the fan. Thus, the forces on the boat would cancel out, and it would not move.
For number 3, I’m pretty sure the answer is yes, he can lift himself up if he’s strong enough, by anecdotal evidence, but I’m not sure how to explain it in terms of Newton’s 3rd law.
For number 4, I suspect that the block would rise closer to the pulley as the man climbs up, so that they are both hanging down the same distance from the pulley. But again, I’m not sure why.
1. You exert a backwards force on the ground, the ground exerts an equal (in magnitude) and opposite (in direction, i.e. forward) force on you. Forward force moves you forwards, backwards force moves THE VERY EARTH ITSELF backwards. Momentum, not speed, is conserved, so the outrageous mass of the planet means it moves backwards an imperceptible amount.
2. Agree with Andrew. At least, for a highschool level question. I bet at a higher level there are all kinds of pedantic real-world circumstances that would produce any effect you want. Hydrofoils cause headaches.
3. The 3rd law actually helps here. This is tough without a diagram… Man pulls rope down, rope pulls man up (effectively reducing the downward force his feet exert on the platform, which is opposed by the normal force). On the other side, rope pulls platform up, platform pulls rope down. The two ‘rope’ forces are always equal, because that’s how tension works. When he pulls down, the ‘equal and opposite’ force pulling up is essentially doubled due to the tension. He’ll only have to lift half the weight of the platform. Hope I didn’t miss any details there.
4. Yeah, the block will rise. If he’s just hanging there (F=Mg), they will balance and nothing will move. If he pulls on the rope to lift himself, he’ll be exerting the force of his mass plus his lifting force (Mg+L) on the rope in the downward direction. The block on the other end exerts only Mg, and since tension is constant on either end, the lifting force L makes the block rise. In the frictionless problem, Andrew is right - the man will move up at half the rate he would on an anchored rope, and the block will stay level with him.
Any inconsistencies raised regarding the 3rd law in numbers 3 and 4 will most likely be accounted for by an increased but balanced downward force on the pulley itself, which is another one of those “it pulls harder on the earth” things, which gets diluted by the earth’s gargantuan mass.
Thanks for taking a shot (it always takes courage!). Between you two you got almost all of the answers right. Check the posted solution for clarifications.