Wednesday 8 August 2012

Newton's law of cooling and more!


A detective is called to the scene of a crime where a dead body has just been found. She arrives on the scene at 10:23 pm and begins her investigation. Immediately, the temperature of the body is taken and is found to be 80o F. The detective checks the programmable thermostat and finds that the room has been kept at a constant 68o F for the past 3 days.

Contour of dead personAfter evidence from the crime scene is collected, the temperature of the body is taken once more and found to be 78.5o F. This last temperature reading was taken exactly one hour after the first one. The next day the detective is asked by another investigator, “What time did our victim die?” Assuming that the victim’s body temperature was normal (98.6o F) prior to death, what is her answer to this question? Newton's Law of Cooling can be used to determine a victim's time of death! Cool, or what!

So, what exactly is Newton's Law Of Cooling?

Newton’s Law of Cooling describes the cooling of a warmer object to the cooler temperature of the environment. Specifically we write this law as,
(t) = Te + (T0 − Te ) e - kt,
where (t) is the temperature of the object at time tTe is the constant temperature of the environment, T0 is the initial temperature of the object, and is a constant that depends on the material properties of the object.

To organize our thinking about this problem, let’s be explicit about what we are trying to solve for. We would like to know the time at which a person died. In particular, we know the investigator arrived on the scene at 10:23 pm, which we will call τ hours after death. At 10:23 (i.e. τ hours after death), the temperature of the body was found to be 80o F. One hour later, τ + 1 hours after death, the body was found to be 78.5o F. Our known constants for this problem are, T= 68F and T= 98.6F.
At what time did our victim die?
Using Newton’s Law of Cooling we have the following two equations,
( τ ) = 80 = 68 + (98.6 − 68) - kτ , 
.....(1)
( τ + 1) = 78.5 = 68 + (98.6 − 68) e - k ( τ + 1),
.....(2)

where τ corresponds to 10:23 and represents the time (in hours) since death.
Simplifying (1) yields, 
12 = 30.6 e - kτ ,
.....(3)
Using the laws of exponents and simplifying (2) yields,
 
10.5 = 30.6 e - kτ e − k.
......(4)
Notice that in (3) and (4), we have two equations and two unknowns (k and τ). Solving for - kτ in (3) gives,
 
e
 

Substituting this value of - kτ in (4) gives,
10.5 = 30.6 *12/30.6e^-k,  e^-k = 10.5/12,  -k = ln(10.5/12),  k approx 0.13353
Now that we have a value for k, we can use this to solve for the remaining unknown, τ. Substituting k ≈ 0.134 into (3) yields,
12/30.6 approx e^-0.13353,  ln (12/30.6) approx -0.13353tau,  tau approx 7.0

We have found τ ≈ 7 hours. Since the detective arrived on scene at 10:23 pm (τ hours after death), the individual must have died 7 hours prior to 10:23 pm or at approximately 3:23 pm.

Thus, the detective would confidently answer the other investigator, “The time of death was approximately 3:23 pm.”
Bingo!




Saturday 19 May 2012

Newton's hair!

Isaac Newton.
Let me tell you all, I had one hell of a time researching about Newton's hair :P

Although I didn't find out much, here are a few things worth writing about it : 

1. He had shoulder length hair, blonde.
2. He never cared to wash it, comb it or even admire it in the mirror.
3. Since he was an alchemist, many locks of his hair was found to have deposits of Mercury on them.
4. His hair turned grey when he was in his 30s (quite obviously)
5. Many people in the late 70s and early 80s sported this hairstyle =)


Brian May, rock guitarist.
Steven Tyler, lead singer, Aerosmith

Monday 14 May 2012

Einstein's success quote

If A is success in life, then A = X+Y+Z. X is work, Y is play and Z is keeping your mouth shut.

Sunday 13 May 2012

Newton's Disc

Newton disc is a disc with segments in rainbow colours. When the disc is rotated, the colors fade to white; In this way Isaac Newton demonstrated that white light is a combination of the seven different colours found in a rainbow. A Newton Disc can be created by painting a disc with the seven different colours. A combination of red, green and blue in the circular disc will yield the same result. This is due to the phenomenon called persistence of vision. It can easily be made at home using a card board piece It was an important discovery as it proves that light is not colourless,but has colour in it which together converge to give a faded white colour which we consider colourless. This property is based on the principles of dispersion of light.

The action and reaction balloon!

Been quite a while since I posted a video, no?
Illustration : Newton's third law of motion.






10 strange facts about Albert Einstein


So you think you know Albert Einstein: the absent-minded genius who gave us the theory of relativity (two of them, in fact, special theory and general theory of relativity), but did you know that Einstein was born with such a large head that his mother thought he was deformed? Or that Einstein had a secret child before he was married?

Read on for more obscure facts about the life of the world’s smartest genius:

1. Einstein Was a Fat Baby with Large Head

When Albert’s mother, Pauline Einstein gave birth to him, she thought that Einstein’s head was so big and misshapen that he was deformed!
As the back of the head seemed much too big, the family initially considered a monstrosity. The physician, however, was able to calm them down and some weeks later the shape of the head was normal. When Albert’s grandmother saw him for the first time she is reported to have muttered continuously "Much too fat, much too fat!" Contrasting all apprehensions Albert grew and developed normally except that he seemed a bit slow."

2. Einstein Had Speech Difficulty as a Child


As a child, Einstein seldom spoke. When he did, he spoke very slowly – indeed, he tried out entire sentences in his head (or muttered them under his breath) until he got them right before he spoke aloud. According to accounts, Einstein did this until he was nine years old. Einstein’s parents were fearful that he was retarded – of course, their fear was completely unfounded!
One interesting anecdote, told by Otto Neugebauer, a historian of science, goes like this:
As he was a late talker, his parents were worried. At last, at the supper table one night, he broke his silence to say, "The soup is too hot."
Greatly relieved, his parents asked why he had never said a word before.
Albert replied, "Because up to now everything was in order."
In his book, Thomas Sowell [wiki] noted that besides Einstein, many brilliant people developed speech relatively late in childhood. He called this condition The Einstein Syndrome.

3. Einstein was Inspired by a Compass

When Einstein was five years old and sick in bed, his father showed him something that sparked his interest in science: a compass.
When Einstein was five years old and ill in bed one day, his father showed him a simple pocket compass. What interested young Einstein was whichever the case was turned, the needle always pointed in the same direction. He thought there must be some force in what was presumed empty space that acted on the compass. This incident, common in many "famous childhoods," was reported persistently in many of the accounts of his life once he gained fame.
4. Einstein Failed his University Entrance Exam

In 1895, at the age of 17, Albert Einstein applied for early admission into the Swiss Federal Polytechnical School (Eidgenössische Technische Hochschule or ETH). He passed the math and science sections of the entrance exam, but failed the rest (history, languages, geography, etc.)! Einstein had to go to a trade school before he retook the exam and was finally admitted to ETH a year later.

5. Einstein had an Illegitimate Child

In the 1980s, Einstein’s private letters revealed something new about the genius: he had an illegitimate daughter with a fellow former student Mileva Marić (whom Einstein later married).
In 1902, a year before their marriage, Mileva gave birth to a daughter named Lieserl, whom Einstein never saw and whose fate remained unknown:
Mileva gave birth to a daughter at her parents’ home in Novi Sad. This was at the end of January, 1902 when Einstein was in Berne. It can be assumed from the content of the letters that birth was difficult. The girl was probably christianised. Her official first name is unknown. In the letters received only the name “Lieserl” can be found.
The further life of Lieserl is even today not totally clear. Michele Zackheim concludes in her book “Einstein’s daughter” that Lieserl was mentally challenged when she was born and lived with Mileva’s family. Furthermore she is convinced that Lieserl died as a result of an infection with scarlet fever in September 1903. From the letters mentioned above it can also be assumed that Lieserl was put up for adoption after her birth.
In a letter from Einstein to Mileva from September 19, 1903, Lieserl was mentioned for the last time. After that nobody knows anything about Lieserl Einstein-Maric.

6. Einstein Became Estranged From His First Wife, then Proposed a Strange "Contract"

After Einstein and Mileva married, they had two sons: Hans Albert and Eduard. Einstein’s academic successes and world travel, however, came at a price – he became estranged from his wife. For a while, the couple tried to work out their problems – Einstein even proposed a strange "contract" for living together with Mileva:
The relationship progressed. Einstein became estranged from his wife. The biography reprints a chilling letter from Einstein to his wife, a proposed "contract" in which they could continue to live together under certain conditions. Indeed that was the heading: "Conditions."
A. You will make sure,
1. that my clothes and laundry are kept in good order;
2. that I will receive my three meals regularly in my room;
3. that my bedroom and study are kept neat, and especially that my desk is left for my use only.

B. You will renounce all personal relations with me insofar as they are not completely necessary for social reasons…
There’s more, including "you will stop talking to me if I request it." She accepted the conditions. He later wrote to her again to make sure she grasped that this was going to be all-business in the future, and that the "personal aspects must be reduced to a tiny remnant." And he vowed, "In return, I assure you of proper comportment on my part, such as I would exercise to any woman as a stranger."

7. Einstein didn't Get Along with His Oldest Son


After the divorce, Einstein’s relationship with his oldest son, Hans Albert, turned rocky. Hans blamed his father for leaving Mileva, and after Einstein won the Nobel Prize and money, for giving Mileva access only to the interest rather than the principal sum of the award – thus making her life that much harder financially.
The row between the father and son was amplified when Einstein strongly objected to Hans Albert marrying Frieda Knecht:
In fact, Einstein opposed Hans’s bride in such a brutal way that it far surpassed the scene that Einstein’s own mother had made about Mileva. It was 1927, and Hans, at age 23, fell in love with an older and – to Einstein – unattractive woman. He damned the union, swearing that Hans’s bride was a scheming woman preying on his son. When all else failed, Einstein begged Hans to not have children, as it would only make the inevitable divorce harder. … (Source: Einstein A to Z by Karen C. Fox and Aries Keck, 2004)
Later, Hans Albert immigrated to the United States became a professor of Hydraulic Engineering at UC Berkeley. Even in the new country, the father and son were apart. When Einstein died, he left very little inheritance to Hans Albert.

To know more about Hans Albert Einstein, click here.


8. Einstein was a Ladies’ Man

Einstein with his second wife and cousin, Elsa 

After Einstein divorced Mileva (his infidelity was listed as one of the reasons for the split), he soon married his cousin Elsa Lowenthal. Actually, Einstein also considered marrying Elsa’s daughter (from her first marriage) Ilse, but she demurred:
Before marrying Elsa, he had considered marrying her daughter, Ilse, instead. According to Overbye, “She (Ilse, who was 18 years younger than Einstein) was not attracted to Albert, she loved him as a father, and she had the good sense not to get involved. But it was Albert’s Woody Allen moment.”
Unlike Mileva, Elsa Einstein’s main concern was to take care of her famous husband. She undoubtedly knew about, and yet tolerated, Einstein’s infidelity and love affairs which were later revealed in his letters:
Previously released letters suggested his marriage in 1903 to his first wife Mileva Maric, mother of his two sons, was miserable. They divorced in 1919, and he soon married his cousin, Elsa. He cheated on her with his secretary, Betty Neumann.
In the new volume of letters released on Monday by Hebrew University in Jerusalem, Einstein described about six women with whom he spent time and from whom he received gifts while being married to Elsa.
Some of the women identified by Einstein include Estella, Ethel, Toni and his "Russian spy lover," Margarita. Others are referred to only by initials, like M. and L.
"It is true that M. followed me (to England) and her chasing after me is getting out of control," he wrote in a letter to Margot in 1931. "Out of all the dames, I am in fact attached only to Mrs. L., who is absolutely harmless and decent."

9. Einstein, the War Pacifist, Urged FDR to Build the Atom Bomb



Re-creation of Einstein and Szilárd signing the famous letter to President Franklin Roosevelt in 1939.

In 1939, alarmed by the rise of Nazi Germany, physicist Leó Szilárd [wiki] convinced Einstein to write a letter to president Franklin Delano Roosevelt warning that Nazi Germany might be conducting research into developing an atomic bomb and urging the United States to develop its own.
The Einstein and Szilárd’s letter was often cited as one of the reasons Roosevelt started the secret Manhattan Project to develop the atom bomb, although later it was revealed that the bombing of Pearl Harbor in 1941 probably did much more than the letter to spur the government.
Although Einstein was a brilliant physicist, the army considered Einstein a security risk and (to Einstein’s relief) did not invite him to help in the project.

10. The Saga of Einstein’s Brain: Pickled in a Jar for 43 Years and Driven Cross Country in a Trunk of a Buick!



After his death in 1955, Einstein’s brain was removed – without permission from his family – by Thomas Stoltz Harvey, the Princeton Hospital pathologist who conducted the autopsy. Harvey took the brain home and kept it in a jar. He was later fired from his job for refusing to relinquish the organ.
Many years later, Harvey, who by then had gotten permission from Hans Albert to study Einstein’s brain, sent slices of Einstein’s brain to various scientists throughout the world. One of these scientists was Marian Diamond of UC Berkeley, who discovered that compared to a normal person, Einstein had significantly more glial cells in the region of the brain that is responsible for synthesizing information.
In another study, Sandra Witelson of McMaster University found that Einstein’s brain lacked a particular "wrinkle" in the brain called the Sylvian fissure. Witelson speculated that this unusual anatomy allowed neurons in Einstein’s brain to communicate better with each other. Other studies had suggested that Einstein’s brain was denser, and that the inferior parietal lobe, which is often associated with mathematical ability, was larger than normal brains.
The saga of Einsteins brain can be quite strange at times: in the early 1990s, Harvey went with freelance writer Michael Paterniti on a cross-country trip to California to meet Einstein’s granddaughter. They drove off from New Jersey in Harvey’s Buick Skylark with Einstein’s brain sloshing inside a jar in the trunk! Paterniti later wrote his experience in the book Driving Mr. Albert: A Trip Across America with Einstein’s Brain
In 1998, the 85-year-old Harvey delivered Einstein’s brain to Dr. Elliot Krauss, the staff pathologist at Princeton University, the position Harvey once held:
… after safeguarding the brain for decades like it was a holy relic — and, to many, it was — he simply, quietly, gave it away to the pathology department at the nearby University Medical Center at Princeton, the university and town where Einstein spent his last two decades.
"Eventually, you get tired of the responsibility of having it. … I did about a year ago," Harvey said, slowly. "I turned the whole thing over last year [in 1998]."





PS : I do not own the images.

Einstein's field equations


The Einstein field equations are the 16 coupled hyperbolic-elliptic nonlinear partial differential equations that describe the gravitational effects produced by a given mass in general relativity. As result of the symmetry of G_(munu) and T_(munu), the actual number of equations reduces to 10, although there are an additional four differential identities (the Bianchi identities) satisfied by G_(munu), one for each coordinate.
The Einstein field equations state that
 G_(munu)=8piT_(munu),
where T_(munu) is the stress-energy tensor, and
 G_(munu)=R_(munu)-1/2g_(munu)R
is the Einstein tensor, with R_(munu) the Ricci curvature tensor and R the scalar curvature.
The opening sequence of the 2003 French film Les Triplettes de Belleville (The Triplets of Belleville) features the Einstein field equations.