Chukky problems

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STAR PRIZE - A FREE COPY OF D GREAT SCIENCE AND ENGINEERING UTILITY SOFTWARE MATHLAB 2009. Everyone knows the worth of Mathlab (500 dollars for the record)

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Chukky problems

  1. 1. CHUKKY PROBLEMSThese problems are geared to show an individual the beauty, the power, the implication and theramification of science and mathematics. The problems are not calculation intensive. Rather the questionsare aimed at highlighting a general misconception among people about some scientific concepts; instigatearguments concerning the result of experiments; illustrate how simple understanding of scientific lawscan solve complex problems; show the brainchild behind some famous mathematical proofs. Some of thequestions also employ more curious scenarios which help to highlight ones level of understanding of ascientific concept.Like every classical problem there is a gift for this challenge. I am awarding a free copy of the renownedengineering and science utility software MATHLAB 2009 which normally costs over 500 dollars forcollege students. All you need to do is answer just one question that applies to you category. Thecategories are listed in front of the questions. You have – elementary school, Junior High School, SeniorHigh School, College, and All Levels. As one question pertinent to your category.Some of the questions may be found online through search engines but it would be of no good to yourselfif you consult such sources. Challenge yourself with the problem, and then after much research, andchallenge, you may consult search engines if you give up. But make a concerted effort to solve theproblems yourself. The whole idea is to educate and not necessarily challenge. So feel free to consult forhelp if need be.If you have any questions, objections, suggestions or ideas about the listed problems below please feelfree to contact me at pjz081@my.utsa.edu. I would be more than excited to respond. Also if you have anyquestions which you feel are challenging or inspiring like the questions below please do contact me andwe can compile and update them into the questions below. Yeah am Mechanical Engineering major so itis pretty much math and physics but please we welcome all sides of science. As long as your questions
  2. 2. can be very challenging please don’t hesitate to drop by my email box. Thank you. Have fun and enjoy.“Science, an interesting epiphany of creation.”Answers to the questions should be posted on Scholars Assembly wall in order for other members of thegroup to learn from your genius and knowledge. If you are not on Scholars Assembly we employ you tojoin us but if you do not want to you can paste your answer on your Facebook wall mentioning my namein it to draw my attention. Since it is mathematics and typing answers would be difficult to write on aFacebook post or to write even in MS word, you may right it in your hand writing and upload it on4shared and then share the link with us. If you have difficulty doing that, just send me the answer as anemail and with your permission I shall publish your answers.I am still pondering on what gift to award our young scientist in secondary and elementary school whoanswer any of these problems. I would be so privileged to have come across such a genius in my life time.If you have any suggestion for what a good academic gift can be please let me know by emailing me.Your humble servant,Chukky
  3. 3. Question 1 – Elementary School/ Junior High School Question Origin - From a Facebook post by Tommy Lash Did you know that the L.C.M of two numbers {x and y} multiplied by the H.C.F of the two numbers {x and y} is equal to the product of x and y. Can you tell me why this is so? That is can you show me why this happens. Note as an elementary school student, I do not expect a formal mathematical proof from you. Just tell me in your own words why this happens? Question 2 – Elementary School/ Junior High School Question Origin – From the tricks my lesson teacher in elementary school taught me. My classmates were always awed at how fast I did this kind of calculations in my head.lol. I had a trick. Did you know that 11 x 14 = 1(1+4)4 = 154; 11 x 16 = 1(1+6)6 = 176;
  4. 4. Like the first question above, I do not need a formal proof I just want to know why this happens. This rule breaks down when it comes to numbers ending with “9” what adjustment is needed to make to correct it.Question 3 – Senior High SchoolQuestion Origin – Friend said, “I thought 0/0 is the same as 1/0”They are clearly not the same. One is called undefined the other is called indeterminate. Only by checking the dictionary meaning of these words one can see what the difference is. But now the question is this, use two examples to illustrate why they are different.Question 4 – Senior High School/CollegeQuestion Origin-Entry Exam Indian Institute of Technology / Putnam CompetitionSolve the following integral- Integrate [x4(1 – x)4]/(1 + x2)from zero to 1? Write your answer in terms of π. Plot a graph of the above equation with a computer or graphing calculator? Shade the area between zero and 1. After you have done all this you should make a very remarkable discovery? What discovery did you make? It took Archimedes years to make the same discovery; it is going to take you only minutes to make this discovery because of your knowledge for calculus which was absent in the time of Archimedes. Aren’t you special? lolQuestion 5 – College
  5. 5. Question Origin – From a mistake in the Engineering textbook Engineering Mathematics by KA Stroud. Yeah, if you get this one, you can write KA Stroud a letter telling him about the mistake of his Maclurin series.1/(1-x) = 1 + x + x2 + x3 + x4 ……… Inserting x=2 gives 1/(1-2)= -1 but if you put it into the Maclurin series you get a huge number that never converges. What is the problem with this series as it is so defined. What is missing?Question 6 – Senior High School/CollegeQuestion Origin – Observation.The next time you enter an examination without a calculator and you want to find the sine of a small angle (in degrees) to approximately 3 significant figures, can you show that sin (x) = 0.0175x for x in degrees. When you proof this you would find out that 0.0175 is only a 3.s.f approximation of a constant. What is that constant?Question 7 – Junior High SchoolQuestion Origin: Famous “Mathemagic Puzzle” by witches and wizard in ancient GreeceWhy is it that you cannot square a circle, that is draw a circle that has the same area as a square. Or why is it that you cannot half the volume of a cube? Lol the day you do any of this please let me know so we can go claim the prize for doing the impossible. lol.Question 8 – College
  6. 6. Question Origin - Mathematics Teacher, November 1987S = 1- 1/2 + 1/3 - 1/4 + 1/5 – 1/6 + 1/7 – 1/8 + 1/9 – 1/10 +1/11 – 1/12 ……….Multiply both sides by 22S = 2 – 1 + 2/3 - 1/2 + 2/5 – 1/3 + 2/7 – 1/4 + 2/9 – 1/5 + 2/11 – 1/6………..Now add the 1st term above to the 2nd, the 3rd to the 6th , the 5th to the 10th, the 7th to the 14th and so on following the pattern of addition given above:Now if you added as above you shall get the following:2S = 1- 1/2 + 1/3 - 1/4 + 1/5 – 1/6 + 1/7 – 1/8 + 1/9 – 1/10 +1/11 – 1/12……….Therefore 2S = S 2 = 1 (This reminds me of love saying that 1 + 1 = 1 not 2.Even the fractions above are in love. lol) So how is this possible. How can 2 = 1 as seen above.Question 9– College (Engineering /Physics)Question Origin – Laplace Equations/ Potential TheoryWhy are the following conditions so crucial to the existence of the fundamental forces of nature like gravity, electromagnetic forces, and even nuclear forces,hence life itself:
  7. 7. Divergence of Vector = Zero; Curl of Vector = Zero. Hint remember that these forces have a field associated with them and whenever you hear field remember flux and whenever you hear flux remember the Prince of Mathematics – Carl Gauss, his flux theory.Question 10 – College (Engineering/Physics)Question Origin: Ampere’s LawThere is a big debate as to whether magnetic force is conservative or not. Well, though this debate exists, how is it that Ampere’s law is obeyed. It requires the satisfaction of the Laplace equation for Ampere’s law to be valid and a condition of the Laplace equation is the conservative nature of the analyzed vector. So what how is it that there is a debate about the conservative nature of the magnetic field. Please do note that the Amperes equation in this case is the pristine form of Ampere’s equation with not Maxwellian correction that is no inclusion of displacement current. To simplify the question, assumes Ampere’s law for a hypothetically infinitely wire carrying a current through it. Use an analysis of a loop around this wire to answer the question.Question 11 – College (Engineering/Physics)Question Origin - FrictionThey say friction is a force that acts in the opposite direction. OK, fine. Let’s imagine we are pulling a box across a rough surface. As we push the box we feel friction pulling back against us right. If I stop pushing, then friction continues to act against the moving truck until it stops right? Now why doesn’t the box stop and start moving in the opposite direction since friction acts in the opposite direction. Or am I wrong can this happen?
  8. 8. Have you ever rolled a ball across the ground and it stops and starts rolling back. Runfor your life if this shouldever happen. lol. Does this remind you about a famous law in physics?Question 12 – College (Engineering/Physics)Question Origin – FrictionHow does a car move? That is, how does a tyre rolling on the ground move forward? What is the force that pushes a tire forward as it rolls on a ground? Remember a free rolling tyre.Question 13 – College (Engineering/Physics)Question Origin – The nature of Force1. This question would help you answer the second. When you do work to an object in aconservative field, where is the energy stored? That is where is the work that you did on theobject stored? In a non-conservative field like imagine pulling a box on a floor in thepresence of friction, where does the work you did on the box go to? What is the difference inthe two cases? Are there really non conservative force? Conservation means that energy donesome where is transformed elsewhere, so is it right to say for example, friction is non-conservative? This highlight the quote by the great 21st century physicist Richard Feynmanwho said no force is non-conservative. All forces are conservative. So can you figure outwhat is wrong with classifying forces as conservative and others as not conservative? A hint:Should it not be called mechanically conservative/non-conservative?2. Isn’t the Work-Energy Theorem a misnomer? If I move a particle from one point to another in a potential field (be it electrical or gravitational) with a uniform speed, I did
  9. 9. work but the Kinetic energy did not change. But I thought the Work Energy Theorem said Work = change in Kinetic energy. Here there was no change in kinetic energy (uniform velocity) but we did work.3. If I get two toy cars and I attach very powerful magnets of opposite polarity into them and then I drive them into each other. Since the magnets are powerful, does the collision cause a deformation on the cart? If you say yes doesn’t that mean that when objects are pulled by the earth’s gravitational fieldthat they undergo deformation? If you still concur yes again does that not then mean that energy is lost when objects fall from one point to another in a gravitational field in the absence of any external influence other than gravity like air, wind or other forces? If this does not make sense to you and you say no then explain why we would not have a deformation. Remember this, for a reaction to occur, entropy must not decrease (2nd law of thermodynamics). Argue for which side is right. Does the answer to this question remind you of the LHC (the Large Hadron Collider). Do you now see what they are trying to do in that never-done-before-experiment, the LHC? That’s the experiment that found neutrino’s moving faster than the speed of light(as they claim)Question 14 – College (Engineering/ Physics)Question Origin: Rotational Mechanics is different from Translational Mechanics but ninety percent of people do not realize this and hence you have errors in physics textbook like, the reason why the earth momentum is conserved is that the sun is too large to move so only the earth moves but that a false answer that seems to be true. So to expose the
  10. 10. falsehood of such an answer, an answer given by many physics textbooks and articles, look at the scenario below: A boy ties a tennis ball with a thin string to winding device. He initially sets the string into a given angular momentum. The experiment takes place in space so no friction and angular momentum is constant. Then the winding device which is fixed to a support is turned on and begins to wind the string in thereby reducing the length of the string. This problem is usually solved by conservation of angular momentum. But here is the question. If you say that momentum is conserved then how is it that this objected changed angular velocity? Hint: Does it take an external torque to change angular velocity? Remember,the rate of change of linear velocity is acceleration (in non-relativistic speeds, that means high speed so we avoid special relativity effects) but is the rate of change of angular velocity angular acceleration (non-relativistic as well)?Question 15 – College (Mathematics)Question Origin – Function ApproximationRemember how Maclurin and Taylor impressed the world by differentiating power seriesparticularly power polynomials to derive polynomial approximations for transcendentalfunctions. Now Fourier says I can even do better and yes he did. Maclurin series can’t work forfunctions whose derivatives are undefined but Fourier’s series can. This is due to the fact that ituses integration as opposed to differentiation to derive its function approximation. Show howFourier series is an integral analogue of its differential counterpart Taylor/Maclurin Series? Canyou also explain why Fourier series is an edge over Taylor/Maclurin Series when it comes toapproximating discontinuous functions? Hint: remember – indefinite integrals.
  11. 11. Question 16 – College (Mathematics)Question Origin: A friend wanted a straight forward proof.Can you show that e is a limit? That is, derive the limit “e”. Show that it is a limit and that thislimit e is associated with the natural antilogarithm.Question 17 (Secondary School)Question Origin: 25 x 25 = (2x3)(5 x 5) = 625 ; 35 x 35 = (3 x 4)(5 x 5) = 1225; 45 x 45 = (4 x5)(5 x 5) = 2025Surprised? It does not end. Watch this 37 x 33 = (3 x 4)(7 x 3) ; 48 x 42 = (4 x 5)(8 x 2) = 2016The question is this; prove the above phenomenon is valid? Use your proof to explain or showwhy this occurs?Question 17 – Open to everyone scientist or non-scientist, to every levelQuestion Origin - From a puzzle from the book “Compendium of Conundrum”A group of birds are in a huge air tight cage, and you put the birds on a weight scale. After youpass some current on the ground so that the birds would get electrocuted if their feet touch theground so they end up flying in the cage (what a cruel experiment!). Now when they are flying inthe cage, would the reading on the weight scale change compared to when the birds were allstanding on the cage. Does it change or not make an argument. Hint; remember the cage is airtight .
  12. 12. Question 18 – Open to all levelsQuestion Origin - A joke by a physics professor about a dummy who says that 64/16 = 4/1because he cancels 6 in the numerator and denominator of the fraction.While this is clearly not the way to do division, lol, can you list other scenarios where this funnyincidence occurs involving numbers less than 100. You can skip cases like 80/10 = 8/1 (cancelthe zeros) or 70/10 = 7/1 (cancel the zero) skip those cases which are very obvious. Hint: Acomputer scientist would find the answer to this problem in no time with a program. Perhaps youcould ask for their help here. Its an easy program anyway.Question 19 – Open to all level involves simple logicQuestion Origin – Legends from famous ancient mathematics challenge.You are asked to mathematically proof that a statement or axiom is mathematically invalid. Whatis the fastest approach to making such a proof? As in you just need to show something and youproof at once that that theory, or axiom is invalid.Question 20 – Senior High SchoolQuestion origin – A funny surprising 2009 WAEC question
  13. 13. Using planks formula E = hf and Einstein’s mass-energy equivalence formula E = mc2 can youderive De – Broglie’s wavelength equation wavelength = h/p where h in the above is planksconstant and p stands for momentum of quantized entity. I would like to remind you that thisconnection between these equations was a coincidence and De Broglie did not just derive hisequation by simply manipulating Einstein’s and Plank’s formulas.Goodluck.

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