Archive for August, 2008

It’s Sarah!

August 29, 2008

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

Physics Friday 35

August 29, 2008

Suppose 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 Pab = πR2·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 = σT4, where σ is the Stephan-Boltzmann constant.
Thus the power emitted by the sphere is this times its surface area:
Pem = 4πR2·σT4

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, 2008

Today, we have another bit of number theory: what are the positive integers n such that n3+1 is prime?

Well, we recall that a3+b3=(a+b)(a2ab+b2). Thus, n3+1=(n+1)(n2n+1).
Now, for n3+1 to be prime, one of those factors must be equal to one. For positive n, n+1>1, and so we have n2n+1=1, which means
which, for positive n, gives only n=1, for n3+1=13+1=2 prime. All other positive n give composite numbers.

Physics Friday 34

August 22, 2008

Suppose 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.
Figure 1
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 vf,

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 vf be the post-collision velocity of mass m, and vF that of mass M. Conservation of momentum gives:

and conservation of energy gives:

Solving the first equation for vf,

Substituting into the energy result:
As vF≠0, we have

(which is double the velocity of the inelastic case).
Note that we do not need vf, as now only mass M is compressing the spring.
We have energy
and so at maximum compression,

Monday Math 33

August 18, 2008

We 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, 2008

Yet 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].

Russia & Georgia Linkage

August 16, 2008

•George Will in The Washington Post: “Russia’s Power Play
  Of particular note is his reminder that Russia has a Security Council veto, so don’t expect action from the UN

•George Will in The Washington Post: “Russia’s Power Play
  Of particular note is his reminder that Russia has a Security Council veto, so don’t expect action from the UN

•Victor Davis Hanson in National Review Online: “Moscow’s Sinister Brilliance

•Michael Ledeen in National Review Online: “The Lights Went Out” & “No Options? Nonsense.
  Why the conflict in Georgia matters to the world, and why America needs to respond

•Kathleen Parker in National Review Online: “Vlad, You’ve Got Mail
  An analysis of American responses with a touch of humor

•Charles Krauthammer in National Review Online: “Making Putin Pay
 Frank J. Gaffney Jr. in National Review Online: “Back to the U.S.S.R.
 In The Wall Street Journal: “Making Putin Pay
  What the US and the West can do to oppose Russia’s actions

•Mona Charen in National Review Online: “The 3 A.M. Phone Call Is Real
  How the candidates have responded and what it says about them

•Former US Ambassador to the UN John Bolton in The Telegraph: “After Russia’s invasion of Georgia, what now for the West?
  A detailed analysis of the new world political situation created by the conflict

•In his blog on National Review Online, David Frum points out one way in which the US in bringing consequences for Russia’s actions: the US-Polish missle defense deal.

The deal had stalled till now over one last lingering issue: the Poles wanted the US to provide as part of the deal a battery of Patriot missiles, operated by US soldiers, to protect them against a Russian attack on their territory. The Russians objected, and the US had accordingly refused to provide them. Indeed, to date, no US combat troops are stationed anywhere in Poland, out of deference to Russian sensibilities.

Suddenly that is about to change. Poland will get its Patriots plus a company of US soldiers. Russia will have to face a missile defense base on its borders. And the US has just served notice in a very painful way that Russian sensibilities suddenly count for a lot less than they used to do.

However, the Russians have responded to this with a threat to nuke Poland.

Howard “YEEAAAAAH” Dean Strikes Again!

August 16, 2008

So 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, 2008

Just go read this Ed Brayton post: That Pesky Constitution