Broken Formulas

I’m aware that the formulas are not displaying at present. The TeX server I was using is no longer around. As time permits, I will be (slowly) editing and reformatting the old posts to use WordPress’ LaTeX plug-in.

Physics Friday 150

Consider a skater of mass m on a smooth skating rink (so that we may ignore friction in this problem) with a cylindrical pillar, of radius R on the rink. A rope (of negligible mass compared to the skater) has one end fastened to the column, and extends straight out tangentially from the column for a length L. If the skater grabs the end of the rope while having a velocity v perpendicular to the rope, and then spirals inward, the rope winding onto the column. If the rope remains straight and taut throughout the spiral, what will the skater’s speed be upon reaching the pillar? In what direction?

Monday Math 149

What is the probability that two independently randomly chosen integers are mutually prime (have no common factor greater than 1)? The probability for four random integers? For n integers in general?

Physics Friday 149

For a block sliding down an inclined surface with angle of incline θ and with coefficient of kinetic friction μ. If we consider the forces, we see that the block can slide down the incline with constant velocity when the net forces are zero. Balancing forces perpendicular to the plane, the normal force is thus . The kinetic friction is thus , while the component of gravity parallel to the plane is thus . We see that these cancel when
, or when .

Now, suppose we have a large incline with , and we start our block sliding with a velocity v₀ in a horizontal direction; that is to say, along the plane in a direction perpendicular to the direction of the slope. What, then, will be the speed a long time later?

Monday Math 148

Find .

Physics Friday 148

A thin bimetallic strip of thickness d is straight at a temperature T₀, with length L₀. The two metals have coefficients of linear thermal expansion α₁ and α₂, with α₂>α₁. If the temperature is raised to a temperature T not significantly greater than T₀, what is the angle θ through which the strip bends?

Total Lunar Eclipse Monday Night

Update Sunday: This post was supposed to be pre-scheduled for tomorrow (Monday), but I hit the wrong button.

Here is NASA’s page on the total lunar eclipse tonight, which should be visible throughout North America. The moment of greatest eclipse will be at 08:16:57 UT (that’s 11:16 PM for my fellow Alaskans and me); the penumbral eclipse begins at 05:29 UT and ends at 11:04 UT

Monday Math 147

Given two non-zero complex numbers z₁ and z₂ such that , show that the arguments of z₁ and z₂ differ by π/2.

Go and Read

Go and read this:
“Finally, A Hamas Leader Admits That Israel Killed Mostly Combatants In Gaza” by Alan M. Dershowitz

The key point:

The Israel Defense Force put the total number of known combatants killed at 709 and the number of known civilian deaths at 295, with 162 (mostly men of fighting age) “unknown.” Such a ratio, if true, would be far better than that achieved by any other nation in a comparable conflict. Not surprisingly, Israel’s enemies initially disputed this ratio and claimed that the number of combatants killed was far lower and the number of civilians far higher. The United Nations, the Goldstone Report, various “human rights” organizations and many in the media automatically rejected Israel’s documented figures, preferring the distorted numbers offered by Hamas’ and other Palestinian sources.

But a statement recently made by a Hamas leader confirms that Israel was correct in claiming that approximately 700 combatants were killed.

Micro-credit

The BBC News reports on the epidemic of suicides and stress-related deaths in southeast India due to pressure arising from inability to repay micro-loans. According to the article, the government of the Indian state of Andhra Pradesh is reporting over eighty suicides in the past few months after defaulting on loans from micro-finance companies. Further, banks have stopped lending to the micro-credit institutions due to fears that large portions of the money loaned may be unrecoverable, in a situation similar to the subprime mortgage crisis.
So, it seems that micro-credit hasn’t lived up to all the talk about how it would help the poor improve their lives