Probability problem from Griffiths QM

[obiesigma]

Howdy everybody. I've browsed this forum many a time but this is my first time posting. Anyways. I'm taking QM in the fall for the second time (the sole reason i didn't graduate this semester), and so I'm going through the griffiths book right now so i can ace this class next time around. I'm sort stuck on this problem (1.13). I'm researching for the university right now and so i could go talk to the prof, but (a) i don't want to look like to much of a kiss-ass by letting him know that i am already studying for his class and (b) I'm semi-terrified of this professor. so voila, here i am. any help would great.

"a needle of length l is dropped at random onto a sheet of paper ruled with parallel lines a distance l apart. what is the probability that the needle will cross a line?"

in the previous problem. i solved for the expectation values of theta, and its <x> projection. I'm thinking the solution will be a combination of the probabilities for theta, and also for where the needle falls on the y-axis (i.e. right on a line or somewhere between two lines) but i can't quite get a handle on the problem. any ideas?

 

[Dick]

If the center of the needle lands on a line the probability it will hit it is 1. If it lands half way between the lines the probability is 0. So figure the probability of hitting the lines as a function of the position of it's center relative to the lines. Then integrate that from 0 to l and divide by l.

 

[DavidWhitbeck]

This is a classic problem, I think you should get a value that is inversely proportional to \pi if you do it right.

I think once you integrate over the center-of-mass business that Dick mentioned you also integrate over it's orientation measured by an angle \theta as you mentioned, and there are only two orientations where the needle still hits the lines (only looking at the case where the center-of-mass is midway between two lines), but dividing by the integral over all variables, including the angle one will get you that silly pi at the end.