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---------------------- 2308 - Nuclear Reactions and Pi from Breadsticks?
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- The probability that a neutron produced by nuclear fission will strike another nucleus upon detonation is 63.7%
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- The probability the throwing breadsticks on the kitchen floor and landing on a tile line is 63.7%.
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- What’s the connection?
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- Be sure the breadsticks are the exact length as the distance between the tile grout. Be sure your throws are totally random. Keep track of the number of times you hit a tile line.
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- If you make 1,000 tosses you will land on a tile line about 636 times. The more tosses you make the closer you will get to the right answer.
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- This probability is determined much the same as with rolling the dice. The probability of rolling a 3 on a die is 1 in six, or 16.7%. Because the die has 6 sides and there is equal probability that any one will be up with a random toss.
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- The breadstick hitting the tile line calculation is a little bit more complicated, but , here is how you do it.
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------------------------- The tile grout lines are exactly 2 feet apart.
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------------------------- The breadsticks are exactly 2 feet long.
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------------------------- Throw the breadsticks over your shoulder in random fashion and count the number of times the stick touches a line.
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------------------------- Mathematically, to determine if a breadstick crosses a line you need to know two things:
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------------------------- (1) How far from the line is to the center point of the breadstick?
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------------------------- (2) Parallel to the line what is the angle of the breadstick?
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- If you know the angle and the center point you can calculate if the breadstick hit’s a line. At any particular angle there are just so many distances of center points that will cross a line. At any particular angle the distance to the line must be the sine of that angle.
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- (Remember, the sine is the opposite side over the hypotenuse of a right triangle formed by the angle.) If we plot the sine of the angle against the angles between 0 and 180 degrees we get a smooth arc from 0 to pi, which is 180 degrees.
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- The top of the curve occurs at one foot and 90 degrees, pi/2, which makes sense. A breadstick at 90 degrees could just touch two lines with the center point at one foot distance from each line.
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- For any angle, if the center’s distance is less than the sine of the angle then it is a hit.
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- If the distance is greater than the sine of the angle than it is a miss.
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- The total number of hits for all the possible angles is the sum of all the distances for all the angles, 0 to 180. This sum becomes the area under the smooth curve plotted for sine angle. In calculus the integral gives you the area under the curve. The integral of sine angle is cosine angle. The cosine of 0 degrees is 1 and the cosine of 180 degrees is (-1)
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-------------------- Integral sine angle from 0 to pi = cosine 0 - cosine pi
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-------------------- Area under curve = 1 - (-1) = 2
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-------------------- Area under curve is 2 .
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- The plot for all possible outcomes of hits and misses is the rectangle one foot high (distance to center point) and pi wide. The area under the rectangle curve is easy to find, 1 times pi = pi.
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----------------------- The ratio of hits to total tosses = 2 / pi
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----------------------- Probability of hitting a line = 63.7 %
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- The probability of a neutron hitting another nucleus in a nuclear fission reaction is 63.7%
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- You can us this same experiment, tossing breadsticks, to determine the number pi. We just did it. Simply solve the equation:
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----------------------------------------- hits / tosses = 2 / pi = 2 / 3.14 = .637
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----------------------------------------- pi = 3.14
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- This experiment was first conducted by a French naturalist, Georges Louis Lelerc Comte de Buffon ( 1707-1788). In 1726 he fought in a duel in Angers and afterwards decided he had better get out of town.
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- He went to London and translated Isaac Newton’s work on calculus in order to learn English. Beginning in 1752 he wrote over 44 volumes of “ Natural History”. He was made a Count by Louis XV. The price of celebrity was high as his son was guillotined during the French Revolution. Buffon (Byoo-fohn) was no buffoon.
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- You can learn more about this experiment by searching the web for “Buffon’s needle”.
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- Other Reviews about Pi:
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- 2137 - The history of Pi
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- 1023 - How Pi is used in the laws of Physics? Euler’s equation.
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- March 13, 2019. 63
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--------------------- Thursday, March 14, 2019 -------------------------
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