None of the regular scheduled posts until 2010.

## Archive for December, 2009

### Hiatus

December 18, 2009### Monday Math 101

December 14, 2009**The Laplace Transform**

Part 2: Convergence

When considering the Laplace transform

,

we must consider the region of convergence; those values of the complex number *s* for which the above integral converges.

A function *f*(*t*) is of exponential order at infinity if and only if there exist real numbers *M* and *a*, and positive real number *T* such that for all *t*>*T*; note then that

for .

Thus, we see that if the function *f*(*t*), t≥0, is piecewise continuous and has exponential order at infinity with , then the Laplace transform is defined for .

In fact, it can be shown that if the Laplace transform converges for some , then it converges for any *s* with , and thus the region of convergence is (if not the entire complex plane) a half-plane of the form , with all or some of the boundary possibly included.

### Physics Friday 101

December 11, 2009If one punctures a small hole in the side of a large reservoir of liquid, at a point below the surface level of the liquid, one will have a jet of liquid from the hole. The lower the hole, the faster this jet will be. If the liquid in the reservoir has a depth of *H*, at what height *h* above the bottom of the reservoir should one place the hole so that the jet gets the most range?

(more…)

### Monday Math 100

December 7, 2009**The Laplace Transform**

Part 1: Introduction

One of the oldest and best known integral transforms is the Laplace transform; it and the related Fourier transform are likely the most commonly used integral transforms. Like the Fourier transform, it transforms a function from one independent variable, the “time domain”, to a second independent variable, corresponding to a sort of “complex frequency.” It has applications in a number of areas, including differential equations, integral equations, and probability theory.

Specifically, for a function *f*(*t*), defined for real *t*≥0, the Laplace transform (more specifically, the “unilateral” or “one-sided Laplace transform) is the function *F*(*s*) defined by

,

where *s* is a complex number.

From the linearity of the integral, we see that the Laplace transform is linear; for functions *f*(*t*) and *g*(*t*), and constants *a* and *b*,

.

The Laplace transform is “unique” in that if two functions *f*(*t*) and *g*(*t*) have the same Laplace transform,

,

then for nonnegative *t*, except possibly at a finite number of separate points.

(More specifically, if , then for all *a*>0).

### Physics Friday 100: Another Classic

December 4, 2009Consider a spool of string with radius *r*, and ends capped by disks of radius *R*>*r*. The spool is placed on its side on a horizontal, so that it can roll on end rims. Suppose one pulls on an unwound end of the string, at some angle *θ* from the underside horizontal, as pictured below.

It’s clear from basic physical intuition that if *θ*=90°, so that one pulls straight up, the spool will tend to unwind, rolling to the left in our picture.

Similarly, if we pull horizontally, *θ*=0°, then the spool will roll to the right, and will wind up the string.

So, then, somewhere between these two angles should be a transition between these behaviors, a critical angle *θ _{c}* where the spool drags without rotating, neither winding nor unwinding. What is this angle?

Solution: