John McCain has chosen Alaska governor Sarah Palin as his running mate!

## Archive for August, 2008

### It’s Sarah!

August 29, 2008### Physics Friday 35

August 29, 2008Suppose we have a spherical blackbody of radius *R*. Now let us shine radiation of intensity *I* on this sphere from one direction. Assuming the sphere is of high enough thermal conductivity to be treated as having a uniform temperature, and neglecting any other heat sources, what will the temperature *T* of the sphere be in equilibrium?

The sphere has some amount of power being absorbed from the incoming radiation; it is in equilibrium when the power it radiates as a blackbody equals this. (Note that if it is above this temperature, it will radiate more power than it absorbs, and will cool toward the equilibrium temperature. Similarly, if it is below this temperature, it absorbs more power than it radiates, and will heat up toward the equilibrium).

The absorbed power is the intensity of the incoming radiation times the area of the cross-section the absorbing surface presents to the radiation. As our object is a sphere, the cross section is a circle of radius *R*, and so the power absorbed is *P _{ab}* = π

*R*

^{2}·

*I*.

Now, we found previously that the power radiated per unit of surface area for a blackbody is given by the Stefan-Boltzmann Law:

*j* = *σT*^{4}, where *σ* is the Stephan-Boltzmann constant.

Thus the power emitted by the sphere is this times its surface area:

*P _{em}* = 4π

*R*

^{2}·

*σT*

^{4}

Setting these equal to find the equilibrium temperature:

.

Note that the radius of our spherical blackbody does not matter, only the intensity of the incoming radiation.

For a numerical example, the intensity of the sun’s radiation at the distance of the earth (solar constant) is approximately 1366 W/m². Using this in our formula, along with *σ* = 5.670400×10^{-8} W·m^{-2}·K^{-4}, we see:

(which is about 6° C). The actual average temperature of the earth (≈288 K, or 15° C) differs from this due to albedo (the planet is not a perfect absorber), atmospheric effects (such as the greenhouse effect), and internal heat generation.

(See also the Wikipedia entry on Effective temperature of a planet).

### Monday Math 34

August 25, 2008Today, we have another bit of number theory: what are the positive integers *n* such that *n*^{3}+1 is prime?

Well, we recall that *a*^{3}+*b*^{3}=(*a*+*b*)(*a*^{2}–*a**b*+*b*^{2}). Thus, *n*^{3}+1=(*n*+1)(*n*^{2}–*n*+1).

Now, for *n*^{3}+1 to be prime, one of those factors must be equal to one. For positive n, *n*+1>1, and so we have *n*^{2}–*n*+1=1, which means

*n*^{2}–*n*=0

*n*(*n*-1)=0

which, for positive *n*, gives only *n*=1, for *n*^{3}+1=1^{3}+1=2 prime. All other positive *n* give composite numbers.

### Physics Friday 34

August 22, 2008Suppose we have a mass *m* moving horizontally with velocity *v* on a frictionless surface. Directly in its path is a mass *M*, backed by a spring at resting length with spring constant *k*.

Now, what will be the maximum distance by which the spring is compressed if the collision between the masses is:

I) perfectly inelastic?

II) perfectly elastic?

I) Inelastic collision means that the masses “stick” when they collide. We use conservation of momentum: the initial momentum is *mv*, and so for post-collision velocity *v _{f}*,

Now, to find the maximum compression of the spring, the simplest method is to use energy conservation: for displacement

*x*from resting length, the potential energy in the spring is . Immediately after the collision, we have only the kinetic energy of the joined masses, so the energy is

.

Maximum compression of the spring occurs when the masses have zero velocity; the kinetic energy is zero, and so the energy is entirely potential, giving us:

.

II) A perfectly elastic collision conserves energy as well as momentum. Let *v _{f}* be the post-collision velocity of mass

*m*, and

*v*that of mass

_{F}*M*. Conservation of momentum gives:

and conservation of energy gives:

Solving the first equation for

*v*,

_{f}Substituting into the energy result:

.

As

*v*≠0, we have

_{F}(which is double the velocity of the inelastic case).

Note that we do not need

*v*, as now only mass

_{f}*M*is compressing the spring.

We have energy

,

and so at maximum compression,

### Monday Math 33

August 18, 2008We know that the harmonic series diverges, but what about the alternating harmonic series ? The key is the Taylor series for the natural logarithm, known as the Mercator series:

, which is valid for -1<*x*≤1. Setting *x*=1 tells us that the alternating harmonic series converges to .

Now, recall the Riemann Zeta Function, which for is given by

.

Suppose we define an analogous function with alternating terms:

.

This series does not have a pole at *s*=1, and in fact, can be defined via analytic continuation to be defined over the entire complex plane. This function is called the Dirichlet eta function.

Now, let us consider the difference of the Riemann zeta and Dirichlet eta functions:

We see that the odd *n* terms cancel, leaving only the even terms:

,

and solving for eta,

which allows us to find exact values for the Dirichlet eta function at positive even integers (and find values for positive odd integers in terms of the zeta function of those integers).

The values for the first few integers are:

### Funny and Wrong

August 16, 2008

[Via Swans on Tea]

### White Mice Cause Cancer

August 16, 2008Yet another “<Insert Product Here> causes cancer in mice” study: Study shows skin creams cause tumours on mice.

Certain commonly available skin creams may cause skin tumours, at least in mice, and experts should be checking to see if they might cause growths in people as well, researchers reported on Thursday.

They found several creams caused skin cancer in the specially bred mice, which had been pre-treated with ultraviolet radiation.

[Via Reuters UK].

### Howard “YEEAAAAAH” Dean Strikes Again!

August 16, 2008So Howard Dean calls the Republican Party the “white” party, while arguing that minorities are ” more successful in the Democratic party.” As Marooned In Marin points out, this is not new for Dean, who called the Republican party “a white Christian party” in 2005, and earlier that year made his infamous “hotel staff” comment to the Congressional Black Caucus. Marooned also points out the actions of ‘tolerant,’ ‘inclusive’ Democrats toward black Republican Michael Steele whan he ran for Lt. Governor of Maryland.

And let’s see;

What party was the president who appointed these:

•First black Secretary of State.

•Second black Secretary of State.

•First black National Security Advisor.

•A black Secretary of Education, a black Secretary of Housing and Urban Development, a hispanic Attorney General, a hispanic Secretary of Housing and Urban Development, and an asian Secretary of Labor.

Since the president in question for all of those was George W. Bush, you tell me.

### Just messed up

August 16, 2008Just go read this Ed Brayton post: That Pesky Constitution