Brainteaser Physics: Challenging Physics Puzzlers

6. Challenges for Your Creativity

? ≥ Ω + √ > cm ? π < = ≈ ≤ AB = 1 Ω = ? 6 Challenges for Your Creativity This chapter contains problems in which the solution requires an element of creativity before the knowledge of physics is applied. PROBLEMS 6.1 Iron Bars Two bars, painted black and with dimensions 1 cm × 2 cm × 10 cm, lie on a table. They look identical but one is made of nonmagnetized iron and the other is made of iron that is permanently magnetized, thus making it a bar magnet. How can you decide which one is a magnet, using nothing but your hands? 6.2 Faulty Balance With a standard set of accurate weights, and with a perfect balance , one can determine the mass of an object. You have the set of weights, but unfortunately the balance is faulty. Without anything placed on the scales there seems to be equilibrium, but you know that the arms of the balance are not of equal length. This has been compensated for by adjusting the weight of the scales. You can still determine the mass of an object. How is it done? 6.3 Greek Geometry Euclid was a Greek mathematician (c. 325–c. 270 BC) whose works dominated geometry for more than 2000 years. Many famous mathematical problems cannot be solved by Euclidean geometry. One of these problems is the trisection of a plane angle, another is the “squaring of the circle” (the construction of a square that has the same area as a given circle, which means finding π), and a third is the “doubling of a square” (finding the side length of a square whose area is twice that of a given square, or the square root of two). In strict Euclidean geometry one is allowed to use only a ruler (without grading, i.e., a straightedge) and a pair of compasses to solve a problem. Our task is even more constrained. You are allowed to use only the ruler—and only to draw straight lines with a pencil. How can you determine the position of the center of mass of a thin L-shaped form by a graphical construction (fig. 6.1)? 128 brainteaser physics Fig. 6.1. Find the center of mass of this figure by a graphical construction, using only a straightedge and a pencil. 6.4 The Sugar Box An open, but full, ordinary cardboard box of lump sugar stands on your kitchen table. How can you determine the static friction factor between the table and the box with the help of a pencil? The error should be less than 15 %. You are not allowed to tilt the table. 6.5 The Catenary In this problem we compare the location of the center of gravity of two hanging objects. First we take two thin, stiff bars, which form a V-shaped figure. It hangs freely from points A and B, as in figure 6.2. The center of gravity (CG) is located half-way down from the horizontal line connecting A and B to the lowest point (C) of the bars. Then we take a chain whose length is equal to the total length of the two bars in the Vshaped figure. When the chain hangs freely from A and B it has the so-called catenary shape. Is the center of gravity of the chain located above, below, or at the same height as for the V-shaped figure? challenges for your creativity 129 Fig. 6.2. Would the center of gravity (CG) lie higher, lower, or at the same level if the two black bars are replaced with a chain of equal length? 6.6 False Impressions Take a close look at the picture (fig. 6.3). Can you find something that is not according to the laws of physics? 6.7 Testing the Hammer The International Association of Athletics Federations (IAAF) gives very detailed rules for the implements used in track and field sports. In hammer throwing, the implement has three parts: a metal head, a wire, and a grip. The head is a sphere, although not quite so because there must be something to attach it to the wire. Rule 191 says about the head: “The centre of gravity shall not be more than 6 mm from the centre of the sphere.” 130 brainteaser physics Fig. 6.3. Find at least three features in the picture that are not according to a law of physics. How can this requirement for the center of mass be easily checked—not in a laboratory but at the competition site? 6.8 Which Way? Figure 6.4 shows a track left by a bicycle. Did the bicycle move to the right or to the left? (The unequal length of the tracks can be explained because the wheels passed through different patches of wet paint before they left the marks on the road.) 6.9 Three Switches In the first floor of a building there are three switches labeled 1, 2, and 3, and with on/off indicated correctly. In the basement corridor there are three ordinary lightbulbs labeled A, B, and C (fig. 6.5). Each switch in the first floor is connected to one (and only one) of the lightbulbs in the basement. There are no other connections between challenges for your creativity 131 Fig. 6.4. Tracks left by a bicycle. In which direction did the bike go? Fig. 6.5. Three switches are connected with three lightbulbs, but how? the three bulbs or between the three switches. It is your task to decide which switch operates bulb A, B, and C, respectively. You are allowed to go down to the basement from the first floor, but only once. After that you must be able to tell how the wiring is done. 6.10 Pulse Beats You have been out jogging, a strenuous exercise. Afterward you take a slow swim in the lake, and just relax. Then you want to see if your pulse is back to its normal value, but you have no watch. Knowing a little bit of physics you soon find out a way to solve your measurement problem. What do you do? 6.11 Fake Energy Statistics The total energy consumption in a country is usually expressed in Mtoe (megatons of oil equivalents). Table 6.1 gives this value for the 25 countries that are currently (2006) members of the EU (European Union). One of the columns refers to actual data (from 2003) and the other column contains fake values. Which is the correct column? SOLUTIONS 6.1 Iron Bars In a bar magnet one end is a magnetic north pole and the other end is a magnetic south pole. The middle of the bar is neutral with respect to magnetic poles. Let one bar remain on the table and hold the other bar vertically so that it touches the middle of the bar on the table (fig. 6.6). If there is an attraction between them, it is the nonmagnetized bar that lies on the table. 132 brainteaser physics Fig. 6.6. A bar magnet lifts a piece of iron, but a piece of iron does not lift a bar magnet, if placed as in the figure. challenges for your creativity 133 Table 6.1. Energy consumption in 2003 by final consumers in the 25 EU countries. The countries are ordered alphabetically according to the official two-letter abbreviation (e.g., BE for Belgium). Country Mtoe Mtoe Belgium 38 62 Czech Republic 25 55 Denmark 15 22 Germany 231 412 Estonia 3 8 Greece 20 51 Spain 90 98 France 158 220 Ireland 11 22 Italy 130 198 Cyprus 2 3 Latvia 4 8 Lithuania 4 9 Luxembourg 4 4 Hungary 18 50 Malta 0.5 0.9 Netherlands 52 97 Austria 26 61 Poland 56 94 Portugal 18 32 Slovenia 5 8 Slovak Republic 10 20 Finland 26 31 Sweden 34 71 United Kingdom 150 219 Mtoe ⫽ megatons of oil equivalents 6.2 Faulty Balance The mass is given by the square root of the results of two weighings, with the unknown mass first on one side and then on the other side of the balance. In mathematical terms, the argument is as follows. The unknown mass is m, and the lengths of the arms of the balance are L1 and L2 . When the unknown mass is placed on the left side, it is balanced by a mass m2 on the other side. When placed on the right side, it is balanced by the mass m1 . We have mL1 = m2 L2 m1 L1 = mL2 Putting L2 = m1 L1 /m from the last equation into the right side of the first equation gives mL1 = m2 (m1 L1 /m) or m2 = m1m2 Even if the balance is obviously faulty and has arms tilting to one side when nothing is placed on the scales, we could use it to determine an unknown mass. Just put weights on one of the scales to restore an apparent equilibrium. We don’t need to know the mass of these weights. Grains of sand, or anything else such as some of the pieces in our set of weights, could be used to get the situation where we can apply the mathematical argument above. Additional challenge. How many pieces of weights, whose masses in grams are given by integer numbers, are needed to measure masses ranging from 1 g to 1 kg, with an error of at most 1 g? The balance 134 brainteaser physics itself is assumed to be very accurate. (See the solution at the end of this chapter.) 6.3 Greek Geometry The L-shaped figure can be divided into two smaller rectangles (fig. 6.7). Each small rectangle has its center of mass (CM) where its diagonals cross. The CM of the two combined rectangles lies on the straight line AB going through the CM of each of the smaller rectangles . But the L-shaped figure can be divided into rectangles in two ways (fig. 6.8). Repeat the construction for the other way. The CM of the L-shaped figure lies where the two straight lines in the two constructions cross. challenges for your creativity 135 Fig. 6.7. The center of mass of the L-shaped figure lies on the line AB. Fig. 6.8. Two ways to divide an L-shaped figure into rectangles In the construction just suggested, the two lines may cross at a rather small angle. That can make it difficult to find the precise position of the crossing point with high accuracy. It would be much better if the two crossing lines make a large angle. We can achieve this with a different construction. Complete the L-shaped figure so that it forms a rectangle. We now have one large and one small rectangle , whose CMs are at their centers. The CM of the L-shaped figure lies on the line through the CM of the two rectangles (fig. 6.9). 6.4 The Sugar Box Push horizontally on the side of the box, with increasing force, until it either starts to slide or to tilt. Which one of these alternatives that happens first depends on how high up on the box one pushes. Try pushing at increasing height h until the box just starts to tilt. With the geometry in figure 6.10, the (unknown) force F then gives the torque Fh counterclockwise around point O. It is balanced by the torque in the opposite direction from the weight Mg. We have Fh = Mga The normal force is N = Mg and hence the fully developed friction force is fN = fMg. The friction force is balanced by the horizontal force F: F = fN = fMg Combining these relations gives the friction factor It remains to determine the ratio of two lengths, a/h. Because the ratio is independent of the length unit used, we may, for instance, measure a and h in units of the size of a piece of lump sugar. f F Mg a h = = 136 brainteaser physics Fig. 6.9. An L-shaped figure plus a rectangle form a new rectangle. 6.5 The Catenary The chain has the lower center of gravity. If you pull the middle of the chain strongly downward, it forms exactly the same shape as the V. The stretched chain and the V then have centers of gravity at the same level. Now release the pull on the chain. It spontaneously returns from the V shape to the catenary shape. This means that the potential energy decreases, that is, the center of gravity of the chain sinks to a lower level. The general shape of a catenary is the same as that of the function This is the form that the cables in a suspension bridge would take if there were no load from the bridge itself. With a load that is evenly distributed along the horizontal direction, the cables form a parabola . cosh( ) x x x = + − e e 2 challenges for your creativity 137 Fig. 6.10. Forces acting on the box when it is just about to tilt counterclockwise 6.6 False Impressions Some of the reflections in the water are not correct, because they show objects that should be hidden behind other objects. For instance , consider part of the chimney, or the root of the tree near the water. Did you find more dubious reflections? The schematic illustration in figure 6.11 shows how an observer may see an object directly but not reflected in the water. Additional challenge. One of the most famous paintings by Diego Velásquez is The Rokeby Venus, which hangs in the National Gallery in London. You may easily find this picture on the web. Take a look at it. What does the woman see in the mirror? (See the solution at the end of this chapter.) 6.7 Testing the Hammer The IAAF rules give an ingenious solution: “It must be possible to balance the head, less handle and wire, on a horizontal sharp-edged circular orifice 12 mm in diameter” (fig. 6.12). 138 brainteaser physics Fig. 6.11. Observer O can see A directly, but not as a reflection. Fig. 6.12. The hammer head should balance on a circular orifice, 12 mm in diameter. Because the head is spherical, but at the same time it is not a perfect homogeneous sphere because of the connection device between the head and the wire, one should orient the head in many ways on the sharp-edged orifice. If it never falls down, the center of gravity lies within 6 mm of the geometrical center of the spherical head, as required by the rules. As an example of how detailed the IAAF rules are, we quote the following from rule 191. “The head shall be of solid iron or other metal not softer than brass or a shell of such metal, filled with lead or other solid material. If a filling is used, this shall be inserted in such manner that it is immovable and that the centre of gravity shall not be more than 6 mm from the centre of the sphere. The wire shall be connected to the head by means of a swivel, which may be either plain or ball bearing.” 6.8 Which Way? The front wheel makes wider and less smooth turns than the rear wheel. It is likely, therefore, that the short track is from the front wheel. Next, we note that the rear wheel is always lined up with the frame of the bicycle, but the front wheel often makes an angle relative to the frame. Furthermore there must be a constant distance between the two points where the front and rear wheels make contact with the road. Draw a tangent to the track of the rear wheel, and in the assumed direction of the motion of the bike. This tangent always coincides with the orientation of the bicycle frame. The distance along the tangent to the point where it crosses the track of the front wheel must be equal to the distance between the wheels, typically about 1 m. Try some tangents in the figure. It is obvious that only a bike moving to the right in the figure can give rise to such tracks. 6.9 Three Switches Let switch 1 first be on and then, after a while, off again. Switch 2 is switched on permanently, and switch 3 is left in the off position. challenges for your creativity 139 When you come to the basement you will see one bright lightbulb. It is connected to switch 2. When you touch the two other bulbs you find that one of them is warm. That bulb is connected to switch 1. Additional challenge. When you come to the basement you find that no lightbulb is lit, but one of them is warm. Can you still figure out the wiring, without returning to the first floor? (See the solution at the end of this chapter.) 6.10 Pulse Beats All jogging shoes have shoestrings. You can make a pendulum with your shoe (or some other object) hanging on the shoestring. Its frequency depends on the length of the string, as described mathematically below. The length itself can be easily determined if you use the fact that the distance between the fingertips of your outstretched hands is very close to your own height. We start with a well-known result for an ideal pendulum (also called a mathematical pendulum). All its mass m is concentrated in one point. If its length is L, the period, that is, the time for a complete oscillation back and forth, is From elementary physics courses it is well known that t depends only on L and on the acceleration of gravity g = 9.8 m/s2, but not on the mass m or on the amplitude of the pendulum, as long as the amplitude is small. An ideal pendulum with L = 1.00 m has t = 2.01 s when g = 9.8 m/s2. It is not difficult to obtain this result by putting numbers into the formula, but if you are a physicist you may have found it useful to remember that “a mathematical pendulum with the period 2 s has the length 1 m.” Of course this is only an approximate result, which rests on the fact that π2 ≈ 9.87 is close to the numerical value of the acceleration of gravity, g = 9.8 m/s2. t = 2p L g 140 brainteaser physics How physicists think. How accurate is a determination of the pulse rate, as described above? No experimental value should be given without an estimation of its accuracy. There are at least three aspects to consider in our case. Perhaps you may have tied the upper end of the shoestring to a tree branch and measured your pulse while it swings. Is it possible to consider the amplitude as small? Second, a jogging shoe hanging on a shoestring is not exactly a mathematical pendulum. Does it matter? Finally, the pendulum motion is damped because of energy losses. How wise is it to take a formula that ignores such effects? To answer the first question we seek a correction in terms of the maximum angular displacement q from the vertical. We expect a series expansion such that the correction to the period of the ideal pendulum is expressed in a power of the small angle q. In advanced textbooks on classical mechanics one may find that (with q in radians) If we let q at the start be less than about 0.17 radians, corresponding to the angle 10°, we can safely neglect the correction due to the finite amplitude. It changes t by less than 0.1 %. Next we consider the fact that a jogging shoe does not have all its mass concentrated in a single point, as required for a mathematical pendulum. Upper and lower limits to this effect are obtained if we let L be the distance to either the highest or the lowest parts of the shoe, when it hangs vertically on the shoestring. Typically, these limits correspond to a variation in L by, say, ±4 %, which gives ±2 % variation in t. The effect of energy loss (damping), leading to a gradual decrease in the angular displacement, is more complicated to analyze. It arises, for example, from the air resistance acting on the shoe as it swings. A more detailed analysis of a damped pendulum (not given here; see “Further Reading”) suggests that its direct effect is not imt q ≈ +     2 1 1 16 2 p L g challenges for your creativity 141 portant in our case. An indirect correction arises because the maximum angular displacement gradually decreases, but we have just noted that the amplitude is not of much concern in our case. Furthermore , if the damping of the swinging shoe is too large, you may want to give the shoe an extra push a few times. If this push is synchronized with the swing, the correction to the measured pulse rate is small compared with several other sources of error, although it is difficult to estimate. Putting it all together, we find that the makeshift clock is remarkably accurate, and should allow you to determine your pulse rate to better than ±2 beats per minute. That should be sufficient for your needs. 6.11 Fake Energy Statistics The column to the right is a fake. The numbers in the two columns are rather close, so it may not be easy to decide which column is a fake just by trying to make an estimation for each country. But if we look at how the nine digits 1, 2, . . . , 9 are distributed among the first digits in each column, we find two quite different patterns. Table 6.2 shows the actual number of entries beginning with 1, 2, etc. in the left (true) and in the right (fake) column, and also the expected number according to Benford’s law (see below). For instance, there should be many more entries beginning with digits 1 to 4 than with digits 6 to 9. Because we have only 25 countries on the table, there will be deviations from the law. An argument for Benford’s law is as follows. Note that entries beginning with 9 or 8 can differ by at most a factor of 9.999 . . . /8.000 . . . = 1.25, while entries beginning with 2 or 1 can differ by at most a factor of 2.999 . . . /1.000 . . . = 3. There is much more room for data in the latter interval. As another argument, suppose that we change the unit (in our case Mtoe) to another energy unit (e.g., TWh). All entries will be multiplied by the same factor. Only a logarithmic distribution for the occurrence of the digits 1 to 9 as the first digit is invariant under such a change of the unit. Of course the 142 brainteaser physics law only holds if we consider a large volume of data, which covers several orders of magnitude. It does not hold, for example, for the length of human lives (be they expressed in years, or months, or days), because normally that length does not vary much. You can test Benford’s law on the web by using a search engine to look for freely chosen numbers, e.g., 130, 560, and 910, and see how many hits you get for these numbers. There should be many more for 130 than for 910. Credit for the rule is often given to the American physicist Frank Benford who, in 1938, published an extensive study. He analyzed numbers on utility bills, lists of prices for various goods, areas of geographical regions, street numbers in the addresses of members of the American Physical Society, and many other collections of data— altogether 20 229 numbers. Almost one third of all numbers have 1 as the first digit. The rule is sometimes called Benford’s law, or the significant-digit law. An interesting application is in the detection of forged data in economic transactions. For instance, if one is freely inventing a large number of expenses, and is not aware of Benford’s law, it is natural to let the first digit in the cost of each item be a number chosen randomly between 1 and 9. The uneven distribution of the value of the first digit was not first discovered by Benford, but had been known at least since the secchallenges for your creativity 143 Table 6.2. Actual number of entries in the left and right columns of table 6.1 and the number expected according to Benford’s law. Digits 1–4 should be more common than 6–9 as the first digit. Value of first digit 1 2 3 4 5 6 7 8 9 Occurrence in left 8 6 3 3 4 0 0 0 1 column (true) Occurrence in right 1 5 3 2 3 2 1 3 5 column (fake) Expected occurrence, 8 5 3 2 2 2 1 1 1 Benford’s law ond half of the nineteenth century. Simon Newcomb (1835–1909) was a professor of mathematics and astronomy at the Johns Hopkins University, with a particular interest in astronomical data. He published his results in the American Journal of Mathematics, the oldest mathematics journal in the Western Hemisphere in continuous publication, which the Johns Hopkins University Press was founded to issue. In a paper from 1881 Newcomb noted that in books of logarithmic tables, the pages for the logarithms of numbers beginning with 1 were more worn than pages for numbers beginning with 9. Tables of logarithms were used to facilitate the calculation of products of numbers, in particular in astronomy. Obviously this was long before even mechanical calculators had been invented. Simon Newcomb was not only a great mathematician, having received many awards, but he also published many popular books on astronomy and even a science fiction novel, His wisdom the defender (1900). ADDITIONAL CHALLENGES 6.2 Faulty Balance With ten weights, having the values (in grams) 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, we can form combinations that cover all integer values of masses from 1 g to 1023 g. With the same set except for the weight of 1 g, we can form all even-numbered masses 2, 4, 6, . . . , 1022 g. If the measurement of an actual mass is always reported as an odd number of grams, we are at most wrong by 1 g. For instance, if the mass is m = 213.7 g, the weighing gives 212 g < m < 214g. We then report the result as 213 g. The required minimum set of weights is obviously related to the digital system of numbers. It contains fewer weights than in a standard set which may have the values (in grams)1, 2, 2, 5, 10, 20, 20, 50, 100, 200, 200, 500, i.e., twelve weights instead of ten, to cover all integer values from 1 to 1000 (actually , to 1110). No doubt a standard set is more convenient in practice , and it requires only two additional weights. But we can also manage with fewer than ten weights. Consider the set 1, 2, 7, 21, 63, 144 brainteaser physics 189, 567, 1701. Each number in the sequence equals two times the sum of the preceding numbers plus 1. Now we can form all integer values from 1 to 2551 with eight weights. For instance, the mass 18 is obtained by putting 21 on one scale and 1+2 on the other. To get 500 we put 567 + 1 + 2 = 570 on one scale and 63 + 7 = 70 on the other. If we are content with only even numbers, as in the binary example above, we can take the sequence 2, 6, 18, 54, 162, 486, 1458. (Can you see how this sequence is formed?) It allows us to weigh all masses from 0 g to 2186 g with an error less than 1 g, and using not more than seven weights. 6.6 False Impressions She sees the painter (or the viewer), but not her own face. There are many famous paintings where the laws of optics are not strictly obeyed. One example is A bar at the Folies-Bergère by Édouard Manet, which hangs in the Courtauld Gallery in Somerset House, London. But physicists should be aware that artwork is not always intended to give a correct visual representation. Just think of paintings by Pablo Picasso. 6.9 Three Switches At least one of the lightbulbs must be broken, but it could not be the warm bulb. You know that the latter is connected to switch 1. Unscrew the warm bulb and let it replace one of the cold bulbs. If it lights up, you know that it is connected to switch 2. Otherwise it is connected to switch 3. challenges for your creativity 145 ...