Henry Greenside's Duke Physics Challenges Duke Physics ChallengesThe following Physics Challenges are for Dukeundergraduates and others to enjoy. The motivation forthese problems comes from Mark Twain's famous quip that"I have never let my schooling interfere with myeducation." and similarly I think physics is too muchfun to restrict just to classrooms, labs, and homeworkassignments.If you would like to propose a solution to a Challenge, talkabout one of the Challenges, or get a hint, please send mee-mail at hsg@phy.duke.edu or dropby my office in the Physics building, Room 097. Pleasealso forward to me any neat problems that you find or inventthat would be fun to present as a challenge.References and links to other collections of physicsproblems and challenges can be found here. Professor Henry Greenside Physics 097 919-660-2548  Challenge 1 Heating of Two Identical BallsYou are given two identical steel balls of radius5 cm. One ball is resting on a table, the other ball ishanging from a string. Both balls are heated (e.g.,with a blow torch) until their radii have increased to thesame value of 5.01 cm. Which ball absorbed more heatand why?Challenge 2 Let Go or Hang On?A painter is high up on a ladder, painting a house, whenunfortunately the ladder starts to fall over from thevertical. Determine which is the less harmful action for thepainter: to let go of the ladder right away and fall to theground, or to hang on to the ladder all the way to theground.Challenge 3 How toknock a bottle over with a sandbag and drinking straw.A heavy 300 kg sandbag one meter tall is hung from aplayground swing with a rope 3 meters long so that thebottom of the sandbag just clears the ground. A bottle isthen placed on the ground a meter away from the sandbag asshown. Explain how to to knock the bottle over with thesandbag if you are given a paper drinking straw but arenot allowed to touch anything (sandbag, rope, bottle,swing) with your body or with the straw.Challenge 4 Running away from killer bees.While walking through an open field on a windy day, youaccidentally step on a nest of killer bees. In whichdirection should you run to save your life? Will you be ableto run fast enough to escape? For this Challenge, assume that the wind is blowing from theeast at 4.5 meters/sec (10 miles/hour) and use thefact that bees have an experimentally measured maximum speedof about 8 meters/sec (18 miles/hour). The fastestrunners can attain 10 meters/sec (23 miles/hour),most people much less than that.Challenge 5 Equilibrationof Two Birthday BalloonsConsider two identical spherical birthday balloons, one ofwhich is inflated to 2/3 its maximum diameter and theother inflated to 1/3 its maximum diameter. Whathappens when the openings of the two balloons are connectedto each other by a straw so that air can flow back and forthbetween the two balloons? Note: This experiment is simple enough that you should try it beforemaking your mind up about what the "obvious" answer is. Challenge 6 Temperature of a kilogramof ice and a kilogram of boiling water?A kilogram of ice at 0oC and a kilogram (liter)of boiling water at 100oC are mixed together in athermally insulated tank. What is the temperature of thewater in the tank after the contents have reachedequilibrium?Challenge 7 Distinguishing two nearlyidentical spheresYou are given two spheres that are identical in size,weight, appearance, and touch but one sphere is hollowwhile the other is solid. (As an example, the solidsphere could be made out of a light wood and the hollowsphere made out of a denser wood, then both spherescarefully painted to look and feel the same.) Usingonly simple items that you might find at home (no fancyequipment, no drills, no X-ray machines), determinewhich sphere is hollow.Challenge 8 Why don't clouds falllike a rock to the ground?As someone living near the beginning of the21st century, can you explain a problem that badlyperplexed the ancient Greeks and Romans (and alsopeople throughout the medieval ages): how come cloudsdon't come crashing down to the ground? After all,clouds are made of water droplets and ice crystalswhich are about 800 times more dense than air,comparable in density to rocks. So why don't cloudsfall like a rock to the ground? To give this problemfocus, propose some specific experiments that you couldcarry out that would help you to discover the answer.Challenge 9 A bird flying betweencolliding trainsTwo trains each traveling at 30 km/hour areapproaching each other on the same straight railroadtrack. When the trains are 30 km apart, a birdresting at the front of one train takes off and fliesat a constant speed of 50 km/hour to the othertrain. As soon as it reaches the other train, itinstantly turns around and flies back to the originaltrain, and keeps repeating this back and forth at thesame constant speed until the trains collide. How farwill the bird have flown at the time of the collision?Challenge 10 Which switch controlsthe desk lamp?A light bulb in a desk lamp is turned on and off byexactly one of three simple switches which are locatedin a remote room such that one can not see the desklamp from the location of the three switches. Explainhow to determine which switch controls the desk lamp ifyou are allowed to flip the switches any number oftimes but are allowed to visit the room with the desklamp only once. You can assume that the on and offpositions of each switch are correctly labeled.Challenge 11 Spherical ThinkingAssuming that the earth is a sphere, where on theearth's surface is it possible for a person to walk onekilometer south, one kilometer east, and one kilometernorth and end up in the exact same place? A hint: there is more than one such place.Challenge 12 Atomic thickness of yoursignatureWhen you write your name on paper using a pencil, you createa thin layer of graphite. Invent and carry out anelementary experiment to estimate how many atoms thick isyour signature.Note: The graphite in a pencil is a pure form of carbonconsisting of many planar sheets of carbon atomsstacked one above the other. The carbon atoms havestrong bonds within a planar sheet and much weakerbonds between the sheets and so one sheet can sliderather easily with respect to an adjacent sheet, whichexplains why graphite is so useful as pencil lead. Thespacing between sheets has been measured by X-raycrystallography to be 0.34 nanometers from whichyou can then determine from your experiment how manyatomic sheets thick is your signature.Challenge 13 Leaning Tower of PizzaAssume that you have a large supply of identical strongsquare pizza boxes of dimension one inch deep by18 inches wide. By stacking these pizza boxes on asturdy table, one on top of the other, how far out intospace can you extend this stack beyond the edge of thetable?Challenge 14 Wrongly rotating wagonwheels of a stagecoachIn watching a cowboy movie, you may have noticed that thewagon wheels of a stagecoach sometimes rotate the wrong way:the stagecoach may be moving left to right across the moviescreen while the spokes of the wheel rotate backwards(counterclockwise). If a movie displays 24 frames persecond, if the wagon wheels are 5 feet tall and haveeight spokes each, and if the horses pull the stagecoachleft to right at 20 miles/hour (32 km/hour), willthe wagon spokes be rotating forward or backwards comparedto the direction of the stagecoach? What will the movieaudience perceive as the angular velocity of the wheels?Challenge 15 Galactic PinballAn indestructible sphere of mass 100 kg is launched byrocket into space. What will its speed be after asufficiently long time? Note: It may be useful for you to know that the mass of astar is of order 1030 kg and the relativespeed of stars in a galaxy is of order 10 km/sec.Challenge 16 A Universal ReflectorConsider three identical square planar mirrors that areglued together to form three adjacent sides of a cubemeeting at a corner, with the mirrored sides all facingtowards each other. Show that these mirrors act as auniversal reflector that sends light back to its source: anylight beam entering this arrangement of mirrors will leaveparallel and opposite to its original direction.If you look into such a corner reflector, what kind of imagewill you see of your face?Note: Such corner reflectors were left on the surfaceof the moon by Apollo astronauts and were used inranging experiments, in which laser beams from anobservatory on Earth were bounced off the surface ofthe Moon and returned to the observatory, with the timeof transit being measured. This enabled the distancefrom the Earth to the Moon to be measured with highaccuracy which has been useful in testing the theory ofgeneral relativity and also for investigating thegeology and origin of the Moon.Challenge 17 Candy-Bar PoweredMarathon Runner A candy bar provides about 300 calories of energy. Bythinking about the physics of running, estimate how manycandy bars a person would have to eat to obtain enoughenergy to run a Boston Marathon of 26 miles and385 yards (42.2 kilometers), if that person weighs65 kg (143 lb) and is 1.7 meters tall (5 feet7 inches).Note 1: A food calorie is a so-called "large calorie", theamount of energy needed to raise one kilogram of water onedegree Celsius at atmospheric pressure, and is equal toabout 4.2 kilojoules.Note 2: Estimating orders of magnitudes of phenomena is afun and important skill and often provides surprisinglyuseful insights into some problems. A famous historicalexample was the order-of-magnitude estimate by Lord Rayleigh(1842-1919) of the lifetime of the sun if it obtained itsheat from chemical means (e.g., burning coal). His estimatedlifetime was orders of magnitude shorter than knowngeological and evolutionary times and so strongly suggestedthat the sun obtained its energy by some unknownnon-chemical mechanism, which we now know to be nuclearfusion.Note 3: Physics 103b at Caltech is devoted to the art of estimating ordersof magnitude. Its web page has problems with solutions thatare lots of fun.Challenge 18 Volume of a Holey Cube Consider a solid silver cube whose side has lengthL=4 cm. If three holes of diameter D=3 cm aredrilled completely through, and perpendicular to, thecenters of all the faces of the cube, what is thevolume V of the remaining metal in the cube?Note: This is not strictly a physics problem but doesrequire the kind of practical mathematical knowledgethat an undergraduate science student should have. Challenge 19 Resistance is FutileConsider an electrical circuit consisting of a cube of 12identical resistors such that each edge of the cube is a1 ohm resistor, and each group of three resistorsmeeting at a vertex are soldered together. Calculate theresistances between nearest neighbor, second-nearestneighbor, and third-nearest neighbor pairs of vertices.Challenge 20 Broken Symmetry GameConsider a circular table (e.g., a bridge table) and a largesupply of identical circular disks that are much smallerthan the table (e.g., checker pieces). Now consider thefollowing simple game: each of two players take turnschoosing a disk and putting it down on the surface of thetable so that the disk lies flat and no disk rests on top ofanother disk.If the first person who is unable to put a disk downloses (because of lack of space), should you go firstor second to win this game?Challenge 21 Deducing the size of theEarth from a lovely sunsetYou are enjoying a Caribbean vacation and happen tohave a stopwatch with you at the beach. As you watchthe sun set over the ocean, you carry out the followingeccentric sequence of events: (1), you lie down on yourstomach in the sand and wait until the top of the sunjust disappears below the horizon; (2), you thenquickly stand up and simultaneously start yourstopwatch. By standing up, a bit of the sun is nowvisible again and (3), you wait until the top of thesun again dips below the horizon, at which point youstop the stopwatch. Knowing this elapsed time, yourheight, and that a day lasts 24 hours, explain howyou can deduce the radius of the Earth. (And next timeyou find yourself watching a sunset at the beach, givethis a try and compare your answer with the known valueof 6400 km.)Challenge 22 Can you trust your heart?According to Daniel Boorstein in his interesting book"The Discoverers" (Random House, 1983), Galileo wasnineteen years old in 1583 when he made theapparent discovery that the period T of a pendulumseemed to be independent of the amplitude A of itsswing (measured in radians, with zero radianscorresponding to the pendulum being directly underneathits support). He was supposedly attending prayers inthe baptistery of the Cathedral of Pisa and wasdistracted by the swinging of an altar lamp, whoseperiod did not seem to change as its amplitude slowlydiminished.In fact, as you hopefully know, the period of a pendulum does depend on the amplitude of theswing, becoming longer as the amplitude becomes larger. Sohere is an interesting historical question: could Galileohave discovered this while in the baptistery? (He diddiscover this later on in his life.) The only clock he wouldhave had available in the baptistery would have been hisheartbeat. Since this is an unreliable clock (one'sheartbeat can speed up or slow down), this raises aninteresting physics question: given an unreliable clock andsome knowledge of what makes it unreliable, how accuratelycan one measure a time interval or difference in timeintervals?Try to do some history of science and determine whetherGalileo could have detected the nonlinear dependence ofperiod on amplitude by just using his heartbeat as aclock. Let's guess that the length L of the lamp'ssupport was L=10 meters and that the amplitude ofmotion was moderate, say A=20 degrees from thevertical.Using the approximate formula "T = 2 Pi Sqrt[L/g] ( 1 +(1/16) A2 + ... )" for the period T of apendulum of amplitude A measured in radians (with L itslength and g=10 meter2/sec the acceleration ofgravity), estimate in seconds the difference in periodbetween an amplitude of 20 degrees and zero degrees.Galileo would have had to detect this small difference intime to realize that the period depended on amplitude.Note: This formula is derived in many textbooks and issurprisingly accurate, even for amplitudes as large as 45degrees (see "Mechanics, 3rd Ed." by L. D. Landau andE. M. Lifshitz (Pergamon Press,1976), Section 11.)Assume that Galileo's heart beat on average 60 beatsper minute but could beat as fast as 70 beats perminute or as slow as 50 beats a minute at any given time. Bycounting heart beats, could Galileo have detected the smalldifference in period between a nearly vertical lamp (A=0)and a swinging lamp with A=20 degrees?Challenge 23 North, South, East, or West,and When? The above picture shows the Sun over eleven successive hoursfrom a location in the arctic. What time was it when the Sunreached its minimum height above the horizon? In whatdirection was the camera facing when the Sun reached itsminimum height?Challenge 24 How sensitive is the humaneye?Some books say that the human eye is so sensitive that itcan perceive the light of a match two miles (3.2 km)away on a dark night. Other books say that the human eye isso sensitive that it can detect as few as five photons(quantized light particles). By using an appropriate orderof magnitude estimate, determine whether these twostatements are consistent with each other.Challenge 25 Law ofreflection for a moving mirror.An elementary fact that people learn about mirrors is thelaw of reflection, that the angle of incidence of a lightbeam striking the mirror (as measured with respect to anormal) equals the angle of reflection.Does this law also hold for a mirror that is moving?Consider a square mirror that is moving at speed v in adirection perpendicular to the mirror. (You can think of themirror as starting in the xy-plane and moving in thepositive z direction of a Cartesian coordinate system.)As the mirror approaches a certain observation point, afriend shines a laser beam of frequency w at the mirror sothat the beam makes an angle A with the normal to themirror. (You can think of the beam as lying in theyz-plane.)What angle and frequency will you measure for the reflectedlight beam? Does the law of reflection still hold?Do your conclusions change if the mirror moves parallel,rather than perpendicular, to its plane (say in they direction if it starts in the xy-plane)?Note: The large mirror of the Hubble space telescope is anexample of a mirror in motion as it orbits the earth. Fromyour analysis, do you think the users of the Hubble have totake into account the motion of the mirror when measuringproperties of its images?Challenge 26 First-ordersingle-variable dynamics are asymptotically boringConsider some quantity y(t) that varies with time t,e.g., the pressure, temperature, mass, voltage, or chemicalconcentration of some system. Show that if this quantityevolves according to a first-order ordinary differentialequation of the form dy/dt=f(y) with f(y) somedifferentiable function, then the asymptotic (nontransient)dynamics of y(t) are boring: y(t) either diverges toinfinity (which is unphysical) or y(t) approaches a constanttime-independent behavior. In particular, no matter howcomplicated the function f(y), the asymptotic behavior cannever be oscillatory.Note 1: In thinking about this problem, try to use aqualitative approach based on the possible signs (negative,zero, or positive) of the function f(y) rather than on anydetailed properties of this function.Note 2: This elementary and neat result suggests thatone needs at least two coupled variables or ahigher-order time derivative to get nontransientnon-constant behavior, e.g., sustained oscillations. Afamous and rather difficult theorem from the turn of thecentury, the Poincare-Bendixson theorem, generalizes yoursingle-variable analysis to two coupled first-orderequations with arbitrary smooth functions of two variables:the only nontransient bounded behavior is either constant orperiodic. For three or more coupled first-order equations,new kinds of nontransient dynamics can occur such asquasiperiodic behavior (multiple oscillations present withfrequencies whose ratios are irrational numbers) or chaoswhich is nontransient bounded dynamics that is neitherperiodic nor quasiperiodic.Challenge 27 Ultimately wrong theoryof everythingA scientist Dr. X of Country Y excitedly holds anews conference and says: "I have finally succeeded infinding the ultimate complete theory of the universe whichwill allow the detailed explanation and prediction of allphenomena from elementary particles to condensed matter togalactic structure to black holes and beyond. It consists ofthe following 23 coupled partial differential equationsinvolving space and time derivatives." At this pointDr. X holds up a 2 meter by 4 meter panelcovered with the exceedingly complex 23 equations sothat the world television audience can be properlyintimidated.Why is this claim obviously wrong?Challenge 28 Vector or not a vector?Consider a smooth vector function F = ( Fx(x,y,z), Fy(x,y,z), Fz(x,y,z) )with three components. If ∂x,∂y, and ∂z denote partialdifferentiation with respect to x, y, and zrespectively, is the triplet ( ∂xFx, ∂yFy, ∂zFz )a vector quantity?Challenge 29 Fission of charged raindropsSmall water drops can sometimes merge into a single biggerdrop (one sees this when rain falls on the windshield of acar). Fortunately, this mechanism seems to have some upperlimit else we might get hit by rain drops a meter or more insize which would be most painful (and probably fatal forsmall creatures). This raises an interesting physicsquestion: what determines the size of rain drops in a storm?Using an elementary knowledge of electrostatics, youcan determine whether electric charge might play a rolein determining the characteristic size of a raindrop. (That electrical charges might play a role issuggested by the occurrence of lightning in manystorms.) Consider a spherical water drop ofradius R carrying an electrical charge Q thatis uniformly distributed over the surface of thedrop. If this drop splits into two smaller sphericaldrops of equal size with each drop having charge Q/2uniformly distributed over their surfaces, show thatelectrostatics favors such a splitting by calculatingthe decrease in electrostatic energy caused by fission.The total surface area of the two equal smaller drops turnsout to be larger than the surface area of the original dropand so, because of surface tension which holds a droptogether, it costs energy to split the originaldrop. If S denotes the surface tension (which has unitsof energy per unit area or Newton/meter), how much energy isneeded to split a drop into two equal smaller drops?Now put your two observations together. For a water raindrop of size R=1 mm and approximate surface tension ofS=0.07 Newton/meter at room temperature, how manyelectrons N would have to be deposited on the drop inorder for the electrostatic energy gained by fission tooffset the energy lost by creating more surface area? Do youthink this would be a reasonable amount of charge toaccumulate by friction as one rain drop bumps againstanother in a rain storm? Is the resulting electric field atthe surface of the drop large in the sense of being close tothe value 30,000 volts/meter at which air breaks down?Note: Find a copy of the Guiness Book of World Records andlook up the weight of the heaviest hailstone ever found (itwas heavy enough to easily kill an elephant!). Evidentlypowerful convection currents in storms can suspend largeweights so that the distribution of rain drop sizes is morelikely determined by stability arguments of the above sortrather than by the largest mass that can be supported in astorm.Challenge 30 Time for a verticalpencil to fall overEstimate the time for a motionless vertical pencil to fallover (1), because of quantum mechanics and (2), because ofthermal fluctuations.Challenge 31 Length of a helicalstringConsider a cylindrical rod of length 12 cm andcircumference 4 cm. Starting at one end of the rod andending up at the other end, a string is wound evenly andexactly four times around the cylinder. What is the lengthof the string?Note: With an appropriate insight, only elementary highschool mathematics is needed to solve this (no calculus, nodifferential geometry).Challenge 32 Time for a marble toroll down and up a kitchen bowl.Consider a hemispherical kitchen bowl of radius R. If amarble is released with zero velocity at one edge of thebowl (a distance R above the kitchen table), howlong will it take for the marble to roll down and then up tothe opposite side of the bowl? For simplicity, assume thatthe marble rolls without slipping.Note: This problem was suggested by Adam Berman, a highschool student.Challenge 33 Critical angle ofrolling for two adjacent cylinders on a tilted board.Consider two cylinders of equal radius and uniform massdensity that are placed on a board so that they are touchingeach other and such that their axes are parallel to thebottom of the board: Now for a single cylinder, as soon as the board is tilted upfrom the horizontal, the cylinder will start to roll. Butfor two cylinders, the tendency for the bottom cylinder toroll is opposed by a friction force arising from the contactwith the upper cylinder. Calculate the critical angle Aabove the horizontal at which the two cylinders will startto roll down the incline.Note 1: Assume that the complex friction forces can bemodeled by the usual simplified rules given in anintroductory physics course. If f and N denote thefriction and normal forces respectively at a contact and ifµ denotes the coefficient of friction, then0 Challenge 53 The Great Snowplow ChaseOn a certain winter day, snow starts to fall at a heavy andsteady rate. Three identical snowplows start plowing thesame road, the first leaving at 12 noon, the secondleaving at 1 pm, and the third leaving at 2 pm. Atsome time later, they all collide. At what time did the snowstart to fall?Note: Assume that the speed of a snowplow is inverselyproportional to the depth of the snow.Challenge 54 A telescope made from arotating mercury mirror.Before recent breakthroughs in telescope design (which allowimages from many small reflecting mirrors to be combined bycomputer into an image corresponding to a single effectivemirror with total area equal to that of the smallermirrors), the largest possible ground-based reflectingtelescope was limited by how large a single plate of glasscould be made and ground into a mirror. However, scientistshad thought of and tried an ingenious alternative, which wasto fill a large cylindrical tank with liquid mercury androtate the tank around its axis. The shiny surface of themercury then deforms into a shape that can be used as alarge and inexpensive telescope mirror.Work out the design of such a rotating mercury mirror. Firstshow that the surface of the mercury will take on the shapeof a paraboloid of revolution. Next, determine the focallength of this paraboloid as a function of the angularrotation frequency w of the tank. Finally, determinethe angular frequency needed to achieve a focal length ofone meter.Is it possible to avoid the drawback that such amercury-based mirror will always point directlyoverhead?Challenge 55 Sinking SubmarinesVersus Floating BalloonsExplain why an inflated balloon (made of a rigid plasticmaterial) will rise to a definite height once it starts torise, while a submarine will always sink to the bottom ofthe ocean once it starts to sink.Challenge 56 The optimal shape for asnowman to reduce melting.A snowman is traditionally made of three balls of snowstacked one above the other. Explain why a sphere is alsothe optimal shape for a snowman body part in the sense that,for a given volume of snow, it will melt the least rapidlyas the weather becomes warm. For example, a snowman made ofcubes, boxes, cylinders, triangular prisms, pyramids, ortetrahedra will melt more rapidly than a traditionalsnowman.For simplicity, assume that the air is uniformly the samewarm temperature, ignore the effects of wind and sunlight,and ignore the fact that the shape will eventually sagbecause of melting and gravity. Challenge 57 Average distance betweentwo random points in a sphere.Show that the average distance between two points that arechosen randomly and uniformly in a sphere of radius 1is 36/35, about 1.029.Most computer languages have a function rand() forgenerating random numbers uniformly in the interval[0,1]. Can you figure out how to use such a generator tocreate randomly and uniformly distributed points in asphere? If so, write a short computer program and confirmyour analytical result by generating many random uniformpairs of points inside the sphere and by averaging thedistances between the pairs of points. Challenge 58 The temperature of a hotspot made from a magnifying glass.You have presumably had the fun of focusing sunlight with amagnifying glass to burn a hole in a piece of paper. Nowthink about this more deeply from a physics point of view:given any arrangment of lenses and reflectors of anyarbitrary size and shape, how hot can you make the beam oflight by focusing it onto a single spot?The sunlight comes from the sun's surface whose temperatureis about 6000 K. If you collect and focus enoughsunlight, could you create a spot hotter than thetemperature of the sun's surface?This problem has an interesting historicalprecedent. Archimedes supposedly recommended that Greekwarriors try to set fire to Roman ships by focusing sunlightwith their shields onto the wood ships. Assuming that theshields were flat and that about 50% of the light isreflected off the shields, estimate how many Greek soldierswould be needed to focus the sunlight and set a Roman shipon fire. A useful piece of data is that a lens of diameter3 cm and focal length 10 cm is capable of burningwood with sunlight. Challenge 59 An inverse rocket and aninverse sprinkler. Consider a cylindrical can of highly compressed gas inouter space. You know that if you puncture a hole in the canso that the gas can leak out, the can will start moving likea rocket, in a direction opposite to that of the leakinggas.Now consider an "inverse" problem in which a cylindrical canis completely empty (has a vacuum) and is inserted into abig tub of water. Also imagine the experiment being done onthe space shuttle so that there is no buoyancy force thatwould push the can to the surface of the tub. The can is nowpunctured at one end so that a jet of water starts to streaminto the can. In what direction will the can move and why?A similar problem: Consider a lawn sprinkler consisting ofthree arms that rotate in a circle as water sprays out. Butnow consider putting the sprinkler at the bottom of a bigtank of water and using a pump to suck water out of thesprinker (so that water now flows into the arms of thesprinkler and out through the hose). In what direction willthe arms turn and why? Challenge 60 Bank shots on anelliptical billiard table.Consider two point balls B1 and B2placed on a mathematical billiard table whose shape is thatof an ellipse, rather than the traditional rectangle. Inwhat direction should one shoot ball B1 so thatit bounces once off the ellipical side wall and hits ballB2? For this problem, ignore the spinning of thebilliard ball.In case you haven't played billiards before, you should knowthat a ball bounces off a wall according to the law ofreflection, i.e., the angle of incidence equals the angle ofreflection as measured with respect to a line normal to thetangent at the point on the wall where the ball bounces. Fora rectangular table, the strategy would be this: drop aperpendicular from ball B1 to a side of the tableand then extend the perpendicular an equal distance beyondthe table to obtain point P1. Draw the linebetween point P1 and ball B2 andidentify the point P2 where this line intersectsthe side of the table. You then want to point your cue stickat point P2 to hit a bank shot that will connectwith ball B2. Challenge 61 Size of smallestasteroid that a person could jump off of.In the not so far future, it may be possible to land anastronaut on an asteroid. Based on how high you can jump onearth, determine the maximize size of a spherical asteroidthat you could jump completely off of. The typical densityof a rocky asteroid is about 3000 kg/m3. Challenge 62 Why does a rotatingpositronium atom live longer?There exists in nature a positively-charged particlecalled the positron, which is the antiparticleof an electron in that it is completely identical (samemass, same amount of electrical charge, same amount ofspin) except that it is positively charged.Experiments show that an electron and positron cancombine into a neutral atom called positronium,which is nearly identical in its properties to ahydrogen atom once the difference in relative mass istaken into account. However, while a hydrogen atom canpersist forever in its ground state, a positronium atomexists only for a short time, about10-10 seconds. The reason is that thereis a finite probability of finding the electron andpositron in the same small region of space in whichcase the two particles can annihilate one other, thepositronium atom disappears, and in its place twohigh-energy photons (gamma rays) are observed.Explain why a rotating positronium atom with large orbitalquantum number l will, on average, live for a longertime than a positronium atom in its ground state beforedisintegrating into two gamma rays. Challenge 63 Unusual lenses ofair and of iron. A scuba-diving archeologist would like to createan underwater magnifying glass out of a flexibleplastic bag filled with air so that she can read thefine print of a sunken Greek ruin. What shape shouldthe bag have to act as a magnifying glass? Sound waves move with different speeds indifferent media and so sound can be refracted andfocused by lenses just like light waves. As animpressive example, the earth's large spherical ironcore acts like a lens that can focus sound wavesemitted by an explosion or earthquake at the earth'ssurface. Using the following information and yourelementary knowledge of optics (in particular, theformula that describes how paraxial rays are refractedby a spherical interface between two media of knownindexes of refraction), estimate the focal length ofthe earth's iron core as a spherical lens embedded inless dense crustal rock. For an additional challenge,work out the details of how sound from an explosion inDurham, North Carolina, will appear as an "image" onthe opposite side of the world. Here are the data: the speed of sound (and so theindex of refraction) varies with depth in the earth ina complicated way, as summarized in a Encyclopedia Britannica figure. For simplicity,assume that the earth's inner iron core of radius3,500 km has a constant sound speed of9 km/sec. Similarly, let's assume that the earth'scrust, which extends from 3,500 to 6,400 km has aconstant sound speed of 11 km/sec. Let's alsoignore the fact that sound waves in a solid medium,unlike light in vacuum, can be longitudinal as well astransverse and these components travel with differentspeeds. The above speeds are for the fasterlongitudinal component. Challenge 64 Does a neutrallybuoyant balloon rise or fall as the temperatureincreases?Consider a balloon filled with helium gas and thenweighted so that it remains motionless in the center ofa sealed box of air at room temperature and atmosphericpressure. If the box is slowly and uniformly warmed sothat the temperature everywhere inside increases by asmall amount, determine whether the balloon will rise,fall, or remain in the same place. Challenge 65 Origin of UnusualRadio NoiseIn 1931, after inventing a sensitive short-wave radioreceiver, an engineer heard an unusual noise on hisreceiver that would appear about the same time each dayand then disappear a little later. In trying to learnmore about this noise, the engineer made carefulmeasurements and discovered that the noise began fourminutes earlier each successive day according to theclock on his wall. What startling conclusion did thesemeasurements imply about the origin of the noise in theradio receiver? Challenge 66 Swimming throughthe air on the International Space Station.Imagine that you are a future tourist on theInternational Space Station and, having forgotten tobuckle yourself into bed at night, you wake up the nextmorning floating freely and weightless in the middle ofyour bedroom chamber. Would it be possible for you to"swim" through the air to get back to your bed? If so,would you use the same kind of swimming strokes as youwould to swim underwater? If you can swim through theair, what would be the order of magnitude of yourmaximum speed?Note: You could always get back to your bed by takingoff your pajamas, wadding them into a ball, and thenthrowing them in a direction opposite to that of yourbed. Conservation of momentum would then give you asmall velocity in the direction of your bed (can youestimate the order of magnitude of this speed?). Buthere the interest lies in the fluid dynamics of a largemass (you) trying to swim through a medium of smallviscosity (air). Challenge 67 Distinguishing aFast Cooler Star From a Slow Warmer StarIntroductory astronomy and physics courses teach theinteresting fact that the color of a remote star can beused to determine how hot the star is. More precisely,these courses teach that the light from a star, whenpassed through a prism or diffraction grating, producesa special rainbow called a "blackbody spectrum" whoseshape (light intensity I plotted as a function ofthe wavelength L of light) is "universal" in thesense that the curve I(L) depends only on thetemperature T of the surface of the star and notat all on the star's chemical composition orsize. Further, the blackbody curve I(L) has asingle peak and the wavelength Lmaxcorresponding to that peak can be used to deduce thetemperature of the star's surface by something called"Wien's law", namelythat T = C / Lmaxwhere C is some universal constant that applies toall hot opaque objects.The discussions in these introductory courses usuallyassume that the star is sitting still in space, but infact most stars are moving toward or away from theEarth, some with a high speed. So consider a remotestar whose surface temperature is T and assumethat the star is moving directly toward the Earth witha speed S. Then something called the Dopplereffect will cause the wavelength of each component ofthe light to become a little shorter("blueshifted"). In particular, thewavelength Lmax corresponding to thepeak of the star's spectrum will be blueshifted and andso an astronomer applying Wien's law to the star'sspectrum will deduce an incorrect higher temperaturefor the surface of the star since in the relationT = C / Lmax, thewavelength in the denominator is a bit smaller than itsactual value.Your challenge: Is there some way for the astronomer todetermine that he or she is looking at a fast coolerstar versus a slow warmer star?Note: The spectra from stars often have dark sharpabsorption lines caused by elements in the outer cooleratmosphere of the star absorbing and then reemittingparts of the blackbody radiation coming from the star'ssurface. It would be straightforward to determine thespeed of the star by measuring how much the wavelengthsof the absorption lines are Doppler shifted compared towavelengths of emission lines of the same elements in alaboratory on Earth. Here the question is how theDoppler effect modifies the blackbody spectrum andwhether, in principle, observations of just theblackbody spectrum can be used to deduce the speedand temperature of a remote glowing object. Challenge 68 The ollie: how does askateboarder get that board off the ground?A basic move in skateboarding is an "ollie", which gets theskateboarder and skateboard high into the air. Since the skateboarder's feet are resting on the skateboardwithout any attachment, use your knowledge of physics (andperhaps of skateboarding) to explain how it is possible to pushdown on the board and so get the board high into the air. Howalso is it possible for the skateboard to stay in contact withthe feet during a jump like this? Challenge 69 Will a neutrally buoyantrelativistic submarine sink or rise?Consider some futuristic submarine that can travel close to thespeed of light while submerged under water and let us assumethat, when the submarine has its propulsion system turned off andis at rest in the water, that its ballast is adjusted so that thesubmarine is neutrally buoyant and so neither sinks norrises. Now assume that the submarine speeds up to close to thespeed of light in some huge ocean, initially traveling parallelto the surface of the ocean. Will the submarine sink, rise, orcontinue to travel parallel to the surface of the ocean?Some thoughts: a fish that is at rest with respect to the oceanand that watches the submarine zoom by will presumably see thesubmarine relativistically contract along its length and so thefish will conclude that the submarine is denser than thesurrounding water and will sink. But sailors in the submarinewill presumably see the water coming toward them at high speedand so they will conclude that the relativistically contractedwater is denser than the submarine and so the submarine shouldrise. So who is right, the fish or the sailors? Challenge 70 Does one have to be quiet inorder not to scare the fish away?Fishermen on the shore of a lake or in a stream often try to bequiet so as not to scare the fish away. Using the fact that thespeed of sound is about 340 m/s in air and about1,500 m/s in water, use Snell's law of refraction todetermine how far back from the shore a 1.7 m tall fishermanwould have to stand so that the sound of the fisherman's voicecould not be heard by any fish in the water. (Assume that thesound does not propagate through the ground.) Show also that ifthe fisherman stands in the water near the shore, then a fishwould be able to the fisherman's voice no matter where the fishis located (although more loudly in some places than others). Challenge 71 Throwing a baseballversus throwing a bowling ball.If you can throw a baseball with a certain maximumspeed, what would be the maximum speed you can throw amore massive object like a bowling ball?Some data: a baseball has an official mass of about0.15 kg (weight of about 5 oz) while 10-pinbowling balls start with a mass of about 3.6 kg(8 lb). The fastest measured baseball pitch had aspeed of 100.9 mph (about 45 m/s). Challenge 72 Doppler shift ornot for co-falling source and detector?A loudspeaker is attached to the bottom end of a 3 mvertical heavy rigid rod and a microphone is attachedto the top of the same rod. If the loudspeaker emits apure tone of frequencyf = 1000 Hz when the rod is atrest, what frequency does the microphone measure as afunction of time if the entire apparatus is droppedfrom a tall tower? Challenge 73 High tide on the Moon.If the Moon were warm and had a water ocean like theEarth, would there be tides on the Moon like there areon Earth? If so, how often would high tide occur? Challenge 74 The economics of a lunar rocket base.President Bush has proposed to build a base on the Moonby 2018 from which future rockets could be launched toMars or other parts of the solar system. Where shouldsuch a rocket base be placed on the Moon, in whatdirection should the rocket be pointed when launched,at what point in the Moon's orbit should the rocket belaunched, and with what minimum speed should the rocketbe launched so that it can escape the Moon-Earthsystem? Challenge 75 Deducing thelocation of heaven from Satan's fall. In the book Dear Professor Einstein: AlbertEinstein's Letters to and from Children edited byAlice Calaprice (Prometheus Books, 2002), a studentJerry from Richmond, Virginia, wrote the followingletter to Einstein in 1952: Dear Sir, I am a high school student and have a problem. My teacher and I were talking about Satan. Of course you know that when he fell from heaven, he fell for nine days, and nine nights, at 32 feet a second and was increasing his speed every second. I was told there was a foluma [formula] to it. I know you don't have time for such little things, but if possible please send me the foluma. Thank you, Jerry It seems that Einstein did not reply to Jerry but thisprovides an opportunity to do some detective work using physics. Assuming that Satan indeed fell from heaven for nine days and nine nights, assuming that Satan's initial speed was zero, and assuming that Satan fell with a constant acceleration of g=9.8 m/s2, deduce how far from Earth heaven must lie and also deduce the speed with which Satan struck the surface of the Earth. Make a more realistic calculation by assuming that the gravitational acceleration g of Satan during his fall was not constant but decreased with increasing distance from Earth according to Newton's universal law of gravity, g=GME/d2, where G is the universal gravitational constant, ME is the mass of the Earth, and d is the distance of Satan to the center of the Earth at a particular moment. (Ignore the fact that Satan would also be acted on by gravitational forces from the Sun and other planets.) What now would be the location of heaven from Earth if Satan fell for nine days and nine nights under these conditions, and with what speed would Satan now strike the Earth's surface? Challenge 76 Which way was thebicycle moving?The picture below shows the tracks made by the wheelsof a bicycle as it was traveling through snow. In whichdirection (left to right or right to left) was thebicycle moving? Which trace corresponds to the rearwheel, which to the front wheel? Challenge 77 Tough balancing actExplain how to arrange ten large identical steel nailsso that they are all supported off the ground by justthe head of an eleventh identical vertical nail. Note: The nails can touch only each other and the headof the vertical nail. You can not use any other itemsin solving this Challenge such as glue or magnets.  Challenge 78 RendezvousYou have been wandering in a desert, crazed with thirst forseveral days, when you see a truck traveling on a straighteast-west road, where the road is south of your location. Inwhat direction should you run to maximize your chance toarrive at the road in time to be picked up by the truck andbe saved?For this Challenge, assume that you run at a constant speedin a straight direction, and that the truck also travelswith a constant speed (whose value you do not know). A hint:you generally do not want to run to the nearest part of theroad.This challenge could also be posed in nautical terms: youare on a small lifeboat when you see off in the distance alarge ship moving in a straight line at a constant speed. Inwhat fixed direction should you paddle as fast as you can(at some constant speed) to maximize your chance ofintercepting the ship so you can be rescued? Top of page Department of Physics var sc_project=646506; var sc_partition=5; var sc_security="57bff993";  |
|
