Problem set 4 Due: Thursday, Nov 5, 1998

1. Consider a chain of 23 identical masses Mj , j=1,...,23 connected to each other by 22 identical springs, with spring constant K and rest length L , and to fixed extremities by two additional springs identical to the other 22. The length of the chain from one fixed extremity to the other is 24 L . If the force on mass Mj is Fj = j(j-24) F find the displacement of each mass from its rest position. Display your results in reader-friendly way (and especially, GRADER-friendly...). (Note: You do not know F, K, L so you need to scale them out).

2. For a Gaussian pdf with mean 1 and standard deviation 3, what is the probability that x will be in the interval [2, 4]?
Using Matlab's randn generate N such random x's (with the correct mean and std) and compare the mean and std of those N numbers to 1 and 3 respectively. Use N=50,2*50,4*50,...,210 *50.

Do these values converge towards the mean and the std of the distribution? If so, can you find out the dependence of the relative difference on N?

3. Calculate the moment of inertia around the y-axis of the "rounded square" rim (or "squared circle") x4 + y4 = R 4 . The mass is uniformly distributed around the rim, so the moment of inertia is the integral around the rim of the distance from the y-axis squared. This is an integral along a curve that reduces to the

integral from 0 to R of 4 x 2 sqrt(1+(y') 2) dx
where y'=dy/dx and y(x)=(R4- x 4 )(1/4) .

This Problem 3 has been cancelled but take a look at the solution .

4. Use Monte-Carlo integration to estimate the volume of material removed from a hemisphere of radius a by a drill of radius a/2 centered at x=a/2, y=z=0 . This is the volume common to the hemisphere x2 + y2 + z 2 < a2, z >0 and the cylinder (x-a/2)2 + y2 < a2/4. Try to get 2 digits of accuracy.

Fabian Waleffe
Wed Oct 28 15:30 CDT 1998