Thursday, August 12, 2010

Giant concerns


Considering the second voyage of Gulliver, i.e. to the land of giants, Brobdingnag, the giants are supposedly 12 times taller than humans. In the case of Lilliputians, they were 12 times shorter. So did Swift get his biology right?
If linear dimension is scaled 12 times, the volume would be 12 cubed, so the mass of such a giant could be easily a thousand times more, whereas the bone area would increase only by a factor of hundred. So each unit area of bone would have to bear a load ten times more than that of in humans. But bones would break at such loads. So as Haldane writes in his essay "On being the right size", each time such giants took a step they would break their bones.
Similarly many other quantities scale with linear size. Even the frequency of the vocal chord will be much lower, so Gulliver would not have been able to communicate with them!

What about trees in Brobdingnag? Would they be 12 times taller than our trees? The reader is encouraged to try it as a problem. Hint: The height of a tree is governed by a balance between gravity and surface tension (which is responsible for capillary action for transport of water and minerals).


What about birds? What would be the size of the largest birds in the land of Brobdingnag. Hint: The flight of a bird is a balance between air drag and gravity. The velocity of take off scales as square root of the wingspan.
Similarly the heat production in the brain would also be ten times more. So the Giants are literally hot-headed!

For details click here
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Further read:
1. On being the right size - J B S Haldane.
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Numerology:
The pair of numbers (5,4),(11,5),(71,7) are called Brown numbers. They satisfy the condition that
for a pair (m,n). Erdos had given a conjecture that these are the only three numbers that satisfy this condition. Can you find other pairs?
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Image courtesy:
1. http://www.theispot.com/whatsnew/2009/12/john-walker-gulliver-s-travels.htm.
2. http://en.wikipedia.org/wiki/List_of_U.S._state_trees.
3. http://www.noelkingsley.com/blog/archives/attitude/.

Thursday, August 5, 2010

Remembering Hiroshima and Nagasaki

World War II still remains fresh in our minds, both for the destruction
that was caused worldwide and for all the new innovations it helped
spawning in terms of science and technology. Though these were initially
aimed at giving a strategic military advantage to those countries who
reaped its benefits, in the years that followed after the war, these
innovative ideas have been put to use for more peaceful means. An object
of fascination, dread and ridicule the world over took birth during these
troubled times during which it was also put to use, the atomic bomb. It
brought the end of the war closer when its effects were experienced by the
speechless masses of people in the cities of Hiroshima and Nagasaki in
Japan. The much debated events occurred on 6th August and 9th August, 1945
respectively. By then, Germany had surrendered and Japan was still
battling against the Allied forces. America, which had earlier in 1941
suffered heavy casualties at Pearl Harbor owing to attacks by the Japanese
were prompted to join into the war efforts on a full time basis. They had
been developing the atomic bomb for some time under the Manhattan project
(1942 - 1946) where the research was directed by Robert Oppenheimer.


The “Little Boy” was dropped on 6th August, 1945 on Hiroshima, Japan
making it the first atomic bomb to be used in war. The sustained fission
of Uranium – 235 was used to create a massive explosion at a height of
around 580 m above the city. The mass of the potentially fissile uranium
used was 68 kg of which only 1 kg effectively took part in the fission
reaction. It was estimated that the energy released in the explosion was
roughly 54 TJ, the equivalent of exploding 13 kT (kilo tons) of TNT.
Working backwards using Einstein's mass-energy relation, the amount of
mass that was actually converted into energy turns out to be only 0.6 g
out of the 1 kg of active fissile material.




There were various components that contributed to the destructive power of
the bomb. The initial destruction was in regions close to the explosion
where people present were witness to the high flux of gamma radiation,
neutrons and extreme heat (~ 4000 K) which led to them being instantly
vapourized. This was followed by a blast wave of very high pressure which
coupled with a sufficient time duration of sustenance and a high wind
velocity resulted in the loss of many more lives and destruction on a
massive scale to buildings within a radius of 1.6 km from the blast site.
At a distance of 1.6 km, it was estimated that the pressure would have
been roughly 34 kPa, still enough to cause structural damage to buildings.
Also, the radiation emanating from fissile material that got scattered
would have also contributed to some long term effects in people such as
radiation sickness, cancer, genetic mutations, etc. which have been
studied over the years.

1. Calculate the amount (critical mass) of Uranium-235 that would be required to sustain a chain reaction.
Hint: For the reaction to continue the number of neutrons produced should be more than the number of neutrons captured in the process. Now fission of each Uranium-235 nucleus creates 3 neutrons, a fraction of which is absorbed or lost through the surface. At equilibrium there is a balance between the amount of neutrons produced in the volume and lost through the surface. The actual value of critical mass depends on the geometry and density of the fissile material. For a pure Uranium-235 sample of spherical geometry, the critical mass is about 30 kg. For details of the calculation for critical mass
click here.

2. Calculate the pressure and the radius of shock wave 10 s after the explosion.

3. What would be temperature at the epicentre few milliseconds after the explosion?

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Numerology:

10662526601 is the only known palindromic number whose cube root (2201) is not a palindrome.
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Image courtesy : wikipedia.

Sunday, August 1, 2010

Some uplifting thoughts !


Jonathan Swift describes the flying island of Laputa in Gulliver's travels in the following way- " The flying or floating island is exactly circular, its diameter 7837 yards, or about four miles and a half, and consequently contains ten thousand acres. It is three hundred yards thick...But the greatest curiosity, upon which the fate of the island depends, is a loadstone of a prodigious size, in shape resembling a weaver's shuttle. It is in length six yards, and in the thickest part at least three yards over...By means of this loadstone, the island is made to rise and fall, and move from one place to another...But it must be observed, that this island cannot move beyond the extent of the dominions below, nor can it rise above the height of four miles."

Was Swift foreseeing a day when objects could be levitated by the use of technology? One can draw some parallels to our current technology that uses a very advanced version of the crude loadstone. Magnetic levitation has become a reality in the recent past, being put to use for transport at very high speeds. Below, we offer a few hints based on the information provided by Swift to see if it is possible today to levitate a island like Laputa based on our current technology.

Steps:
1) Calculate the magnetic field strength necessary to keep the island afloat?
When magnetic force balances the island against gravity we can equate the total magnetic energy to the gravitational potential energy .

where B represents the magnetic field strength in a volume V, m is the mass of the island ( assume the density of Laputa to be same as of Earth), g is the acceleration due to gravity and h is the height to which the island is levitated.
In comaparision the magnetic field strength of the Earth is typically 0.5 Gauss.


Suppose we want to levitate a house with a giant balloon (like in the movie Up), what would be the size of the balloon required? We can use the following expression


where rho is the density of gas, V is the volume of the balloon and m is the mass of the house. Compare with the volume of the largest gas balloon launched which is about 60 million cubic feet.

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Numerology:

The Euler-Mascheroni constant is the limiting difference between the harmonic series and natural logarithm. It is denoted by the greek letter gamma.


There is a prime number version of the Euler-Mascheroni constant known as the Meissel Mertens constant which is the difference between harmonic series and natural logarithm of natural logarithm.


where p is a prime number less than or equal to n.
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Image courtesy:
1. http://www.wisdomportal.com/Levitation/LevitationLiterature.html
2. http://www.popbytes.com/archive/film/reviews/