Problems, problems, and problems

These problems are a collection of problems on topics that interest me , at the university level, many taken from standard university texts. As the solutions are my own, so then will be any errors. Errors brought to my attention will be posted with the appropriate credit given.

One must learn by doing the thing; for though you know it, you have no certainty until you try.  -- Sophocles

You should keep in mind no names, nor numbers, nor isolated incidents, not even results, but only methods. The method is plastic. It is applicable in every situation.  -- Emanuel Lasker, from Lasker's Manual of Chess

Mechanics

Terminal velocity

Compute the terminal velocity of a projectile thrown upwards with initial velocity v ...

Inertia Tensor

Calculate the moments of Inertia for a homogeneous cone of mass M whose height is H and whose base has a radius R. Choose the x3 axis along the axis of symmetry of the cone. Choose the origin at the apex of the cone, and calculate the elements of the inertia tensor. Make a transformation  such that the center of mass of the cone becomes the origin, and find the principal moments of inertia

Probability

Typing monkey

Calculate the probability a monkey randomly typing will produce a Shakespearean...

Electricity and Magnetism

Charged Spherical
Shell

A spherical shell of radius R carries a uniform charge sigma on the northern hemisphere and a uniform surface charge -sigma on the southern hemisphere. Find the potential Phi inside and outside the sphere, calculating the coefficients of the series solution explicitly up to A8 and B8

B field due to solenoid
carrying current I and
N turns per unit length

For a solenoid wound with N turns per unit length and carrying a current I, show that the magetic-flux density at the point P is given for N -> infinity by 

...
 where the angles are defined in the figure. (From Jackson 5.2(a),Classical Electrodynamics, 2nd Edition)

Magnetic force on a long,
straight bar with magnet-
ization M.
(Solution coming...)

Show that in general a long, straight bar of uniform cross-sectional area A with uniform
lengthwise magnetization M, when placed with its flat end against an infinitely permeable flat surface, adheres with a force given by, approximately,
                   
(From Jackson 5.14,Classical Electrodynamics, 2nd Edition)

Quantum Mechanics

Commutator of a 
function of p with r

Baker-Hausdorff
problem

Shifted Wave Function