Physically Incorrect

While yesterday’s puzzle still remains unsolved, I thought I’d share this Youtube video having to do with an earlier puzzle that explained why birds don’t get electrocuted when sitting on high voltage wires:

While not directly related to the birds puzzle, it’s still fascinating to watch (note, though, that the two wires the guy sits on are at the same potential; the far away wires are at a different potential).

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?

1. 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?
2. 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?
   
3. 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.
   
4. 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.

Level of Difficulty: Highschool

After Reyni’s fiendish car parking problem, we’re back at physics problems. You all know what a grandfather clock looks like:

These clocks rely on the pendulum within them to keep time. Treating it for the moment as an ideal simple pendulum, it swings under the force of gravity with a period, T, given by:

g is the gravitational constant on Earth (g ~ 9.8 meters / sec^2) and l is the pendulum’s length.

How would you get this pendulum clock to work in deep space, without gravity to make the pendulum swing?

EDITED, 21/June/2009: hats off to our reader Dhruv for his solution. You can read it and my follow-up in the comments section.

Level of Difficulty: Advanced Undergraduate

You’re observing a parking lot of length L, initially empty (this is a one-dimensional parking lot, mind you). Cars start coming in sequentially, one after the other. Each car has width W, and parks randomly with equal probability somewhere in the space that’s available. Eventually the parking lot will fill up. The question is, when space runs out and the parking lot gets filled up, what percentage of the parking lot will be filled on average? I’ll call that the filling factor.

Note that the parking lot is “continuous”. There are no marked parking spaces - every car that parks does so with equal probability at each available point. Also, this is not a problem in psychology. Forget about the way humans behave when they park. The probabilities here are totally random (a.k.a. equiprobable). Here’s a little illustration to make things clearer:

Now, a word of warning. This puzzle is very difficult to solve exactly, so I’m not going to ask you to do so, and you shouldn’t be hard on yourself for not solving it, regardless of your skill level. Instead, I’m going to ask you to estimate the filling factor. For example, to get you started: can you bound the filling factor from above? From below? Be creative! I’m more interested in your reasoning than the actual answer (I already know it).

Good luck and have fun!

EDITED, 21/June/2009: the comments now contain the actual answer, links to a detailed solution (from a paper, in pdf) and  further discussions in other sites. Still, the problem of estimating the result correctly remains open, so if you happen to stumble by this page do take a stab at it. Trying to approximate the solution is, I think, more interesting than deriving the actual solution itself!

Level of Difficulty: Highschool

A 10-kg block is hung from a rope. The rope is connected to a (massless) digital scale which displays the attached weight, as shown in (A).

Next, a pulley is attached to the block via a piece of rope,  and a second 10-kg block is hung from the pulley, as shown in (B). The rope’s other end is attached to the (immobile) floor. What will the scale read now?

All ropes are massless, as is the pulley. Neglect friction.

EDITED, 19/June/2009: a solution has been posted in the comments section by our reader DU; I’ll be posting a more detailed solution later tomorrow.

EDITED, 20/June/2009: click here for a detailed solution.

Level of Difficulty: Undergraduate

Ah, another missing energy puzzle! I love those.

It is well known that wave phenomena that are described by the wave equation obey a superposition principle: different disturbances simply add up. Furthermore, disturbances travel forward with a constant velocity while retaining their shape (assuming no dissipative forces are present, which we will assume).

Consider two waves having identical shapes but opposite phases, traveling in opposite directions on a string:

Two waves traveling towards each other interfere destructively

As shown in the illustration, at some point the waves will add up destructively (after which they will re-emerge). The question is: at the moment they add up destructively it seems the disturbances cancel out; where does the energy “go”?

EDITED, 19/June/2009: a solution has been posted here. Hats off to our reader DU for spotting the missing energy. Note the posted solution contains both a physical and mathematical discussions. The math can, of course, be ignored by those not into it.

Level of Difficulty: Highschool

You are in the middle of a field surrounded by a circular fence of radius R. Your objective is to escape by reaching the fence. However, life isn’t so rosy. Patrolling the perimeter is a guard dog.

At every instant the dog tries to minimize the distance between the point nearest to you on the fence and him (however, the dog is confined to the perimeter):

The bad news is that, while your maximal speed is v, the dog’s maximal speed is 4v. This means that you can’t just make a straight dash for it (can you see why?). You must come up with an escape strategy. What will it be?

(Note: for the record, dogs make one of the most loyal and affectionate pets there are and if you ever come across an ill-tempered one, the owner is to blame.)

EDITED, 18/June/2009: a solution has been posted here. Hats off to our reader DU for his solution in the comments section.

Level of Difficulty: Highschool

This week, six puzzles with matchsticks! I’m going to take a break from writing for a change and let this little Youtube video I’ve created explain it all:

In case you’ll need them, solutions will be published in a couple of days, so check back soon!

EDITED, 17/June/2009: a solution has been published here.

Level of Difficulty: Highschool

Here’s a classic puzzle without which I couldn’t call this a puzzle-blog.

Here is a cube with a resistor on each edge. All the resistors have the same resistance R. What is the equivalent resistance between points A and B?

A Resistor Cube

For dessert, here’s a variant that’s not often seen and requires a bit more work: can you compute the effective resistance between points A and B in the cube below? As before, all edges have the same resistance R:

Another Resistor Cube

EDITED, 16/June/2009: a detailed solution has been posted here. Hats off to our reader R4mOn for his solution in the comments section!

Level of Difficulty: Highschool

Quick, answer true or false to the following without doing any lengthy calculations. Justify your answers:

1. Pump air out of a sealed jar. The speed of sound will increase. True or false?
2. Heat up the sealed jar. The speed of sound in the enclosed air will increase. True or false?
3. The air enclosed in a typical room weighs more than a 5 kilo weight. True or false?
4. A sound wave transitions from air to helium. Its frequency increases. True or false?

EDITED, 14/June/2009: I’d just like to make it clear that in questions 1 and 2, some of the air is pumped out and some is kept in - enough for sound waves to propagate.

EDITED, 15/June/2009: the solutions have been posted in the comments section.