Contents
Answers
Part 1: One-way processes
1 If the reversed event were a real event, how on earth would the pieces contrive to assemble themselves correctly without help? And how would the necessary energy to lift each piece by the proper amount be gathered together?
2a The sketches were made when the pendulum was at one or other of its maximum displacements, though it is not possible to be sure this is so from inspection of the drawings alone. If you care to guess that it is so, the events are in the proper sequence, because damping is reducing the amplitude of the swings.
2b The sketches are in reverse sequence.
2c It is not possible to tell whether the sequence is in the proper order, or is reversed.
2d This is the correct sequence. The water in the middle can would never, unaided, become hotter at the expense of the water around it, so the drawing containing the larger temperature difference must be the earlier one.
2e In the correct sequence, supposing that the ball is slowing down as it runs from left to right.
3 The energy boxes might look like figure 91.
Figure 91
4Perhaps one might hazard the guess that when a process involves energy being spread out among more objects or among more molecules, its reverse would look absurd. For a good air track, the energy of the vehicle stays almost fixed, and very little leaks away, to become spread out among the molecules of the vehicle and track, making these bodies a shade warmer.
Interestingly, if one started with a number of vehicles at rest on an air track, and sent one vehicle along to collide with them, a film of the resulting motion would also look implausible, even if every vehicle moved without any loss of energy to the track, and all collisions were elastic. Again, a process involving the sharing of energy among many objects is one-way in nature.
5 It is more likely that everyone in a class of three would happen to make the same choice at the same time than that everyone in a class of forty would do so. The chance that any one person will choose to go to the cinema might be the same, but the combined chance that many will do so is smaller than the combined chance that few will do so. If there are, say, six equally favoured ways of spending the evening, one person has six choices. Three students have 6 × 6 × 6 patterns of evening entertainment, while forty have as many as 640 all told.
You would be right to think that the example is artificial. Ways of spending one's time are not equally favoured. If there is a film everyone wants to see, or if there is an organized party, the analysis fails.
6 The Brownian motion looks just as good in reverse as it does in the forward direction. The motion of the jiggling particle (a smokey ash particle in air, for example) is unpredictable in either direction.
7 An electric motor, run in reverse as a dynamo, would be an example.
8 Figure 92 illustrates how a proportion of the energy transformed in the battery goes, via the motor, to lifting a load. That part of the whole process which results in the load being lifted would look plausible in reverse, but to the extent to which the motor becomes warm (current flowing in the resistance of its rotor, for example) the process is one-way.
Figure 92
9 Figure 93 illustrates how all the energy transformed in the battery ultimately goes to warming up the surroundings. The reverse process, with the lamp gathering in energy from around itself, and charging the battery, would be absurd.
Figure 93
Note concerning the one-way or two-way nature of a reaction in an electric cell
See Part Six, example 8, for a fuller discussion of the energy-transforming processes in a cell. Whether the cell action warms or cools its surroundings depends on what happens inside the cell. The passage of a current through the cell always delivers energy which goes to warming up the cell and its surroundings, since the cell has a finite resistance, but the cell action may also involve removing some energy from the surroundings. The answer to question 9 is, overall, unaffected, but the answer to question 8 may involve some subtleties.
Part 2: The fuel resources of the Earth
1 Almost 200 years.
2 Just after the year 2000, say 2005. If you were born in 1955 you would be 50 years old.
3 About the year 2020. Children born in the 1970s will be under 50.
4 A growth with a fixed annual percentage increase is exponential in form.
5 In 1969 there were about 11 million private passenger cars in Great Britain. You might have guessed at such a figure by supposing that the population of about 50 million falls into families of rather less than five people each, with rather fewer than one car per family.
6 A car engine can deliver somewhere around 60 or 70 horsepower, say 50 kW. You might reach such a guess by supposing that a car engine is equivalent to a good number of vacuum cleaner electric motors, for example. Or you might know that a car can accelerate to, say, 50 kilometres per hour in a time of the order of 10 seconds, and estimate the power from the rate of increase of kinetic energy, having estimated the mass of a car. Another way would be to reflect that the heater of a car, using perhaps one or two per cent of the engine power, warms the inside of the car not much less quickly than a one kilowatt electric fire would do.
Overall, one might put the power installed in cars at some 500 000 megawatts (5x 1011 W).
7 A power station delivers at least 100 megawatts, often more. You might guess that there are at least 100 power stations in Britain. In 1967-8 there were 216 power stations, with a total capacity of nearly 42 000 megawatts ( 4.2 × 1010 W), an average of just under 200 megawatts each.
Perhaps surprisingly, the power installed in the nation's cars (see question 6) is a whole order of magnitude greater than that installed in its power stations.
8 One way of guessing the total usage of petrol would be to reckon that a car uses some four or five gallons (about 25 litres) of petrol a week, that is, about 20 kg. In a year this amounts to about 1000 kg (1 tonne), so all the cars in the country (see question 5) use about 10 million tonnes. In energy terms, this is between 4 × 1017 and 5 × 1017 J. It represents about 1/25 of all the energy consumed in Britain in a year.
9, 10 The power station is beside a large river. The warming of river water by power stations can have considerable consequences for the animal and vegetable life the rivers can support. Ultimately, the energy used by men warms the Earth until it radiates at a rate sufficient to get rid of the energy delivered to it. At present, the world as a whole is using energy at about 1/10 000 of the rate at which energy arrives from the Sun. But if everyone in the world had the energy standard of living of North America (over five times the world average), the fraction would increase to nearer 1/2000. Because a temperature rise of the Earth of only a few degrees would be enough to flood much of the land area as the polar ice-caps melted, this fraction cannot become more than a small one without serious consequences.
11 It would be very difficult to work all day at a power of more than a hundred watts or so. Thus, a day's work comes to less than one kilowatt hour, which is the unit in which electrical energy is charged to consumers. A kilowatt hour costs about 1p, or less.
12, 13 Less than 1 p (see question 11). The cost of electrical energy is much less than the cost of the food a labourer needs, let alone the cost of wages, so in effect, electrical energy is slave labour. Men as energy-producers alone are far more expensive. In India, for example, this has very direct consequences: it may be cheaper to dig an irrigation canal with machines, but it may be better, in terms of giving people employment, to dig it by manpower.
Part 3: Chance and diffusion
1 No. The marbles could sort themselves out, but unsorted patterns are more likely than sorted ones, because there are more of them.
2 There are very many ways of arranging the marbles so that they are muddled and well mixed.
3 There are fewer arrangements of marbles that one would call well sorted than there are arrangements one would call well mixed. One could argue this from the fact that, given a sorted collection, most changes make it less sorted, but given a mixed arrangement, one has to choose changes with care to move to a better sorted arrangement.
4 Yes.
5 About two times in six.
6 Because there are only two vertically sorted ways as against four not vertically sorted.
7 No.
8 4.88.
9 Yes.
10 One must never expect things to average out too well in the short run.
In the long run, about one picture in sixteen should show the pucks all in one particular half. In thirty-odd frames, it is not too surprising that the pucks never appear together in the lefthand half, nor that they appear three times in the righthand half. In three hundred frames, it would be extremely odd if there were no occasions when all the pucks were in either half.
11 One way would be to toss a coin once for each counter, putting the counter in the righthand or lefthand half according to whether the coin showed heads or tails.
12 No. The average behaviour is with the counters equally spread between the two halves.
13 On the first throw, a counter must move from the full half to the empty half.
14, 15 Five moves out of the six possible moves. The other move will bring the first-moved counter back again.
16 One person in every six.
17 1/W.
18 lg 2100 = 100 lg 2 ≈ 30.
Therefore, 2100 ≈ 1030.
19 1024 seconds.
20 About three million Universe-lifetimes.
Part 4: Thermal equilibrium, temperature, and chance
1 No. The temperatures will have become equal.
2 From the hot water.
3 Because the sandwich is cooler than the hot water.
4 Because it is cooled by the cold can.
5 Not for long. The sandwich would become nearly as hot as the hot can.
6 Because the cold can must keep the sandwich cooler than the temperature of the hot can. Energy going into the cold can is diverted from the motor.
7 No, some goes to the cold can.
8 Energy would seem to come from a dynamo and, going to the sandwich, would seem to appear in one can which became hotter. At the same time some more energy would seem to go from the other can, which would seem to grow cooler. This energy would also seem to be arriving in the can which was growing hotter.
9 That if put in contact, neither will grow hotter or cooler.
10 No, and no again! Your bath water is much more expensive to heat than the hot water in a cup of coffee. But the second may well be the hotter. To bring both lots of water to the same temperature requires quite different amounts of energy. The amount of energy needed to warm something by a certain amount depends on what it is, and on how much of it there is.
11 It will oscillate.
12 Yes. Energy would tend to be transferred from one to the others.
13 It would soon follow the others.
14 Record A would be different; B would be much the same.
15 Always more atoms with n quanta than with n+1.
16 Not the same.
17 The ratios of the numbers of atoms in adjacent energy levels show some very rough tendency to be the same, though the evidence that the ratio is constant could hardly be called convincing.
18 Aspect b. The distribution describes the relative numbers of atoms in the various levels.
19a No.
19b Yes, more or less. Pretty well all games will end up with more atoms in each level than in any higher level. The overall behaviour can be predicted, but not the detailed pattern within that overall behaviour. If you tossed a coin a thousand times, you could predict about how many heads you would get, but not which tosses would give heads.
20 No. It was told to move quanta completely at random.
21 No. The moves were quite random. That is a way of saying they are unpredictable.
22 No. It was told to move quanta completely at random.
23 Neither atoms with many quanta, nor atoms with few quanta, come in for special favour by the random choice of atoms. But if there are many atoms with a certain energy, a lot of random choices will chance to fall on one or other of them.
24a No.
24b Yes, there are so many of them that random choice will tend to hit upon one.
24c Yes, for the same reason.
24d Yes, for the reasons given in a and b combined.
24e Yes. Despite not knowing where to go, the random moves go somewhere definite.
25 Yes.
26 No.
27 Different atoms have each particular possible energy.
28 There is no real change in the numbers of atoms with each particular energy. The distribution is much the same.
29 It ends up with the same number of atoms having each particular energy.
30 It fluctuated up and down around the steady general shape, which did not alter.
31 It would go back to a distribution such as the one shown in figure 40.
This is a puzzling problem. If a series of random numbers choosing sites are random one way, are they not random in reverse order? Yes, if the reversed order numbers were used starting at figure 39, the effect would be to produce the distribution of figure 40, as in all other instances in these films. But applied to figure 40, with the quanta just as they are placed in that figure (not just with the same distribution) this is the one set of random choices that leads back to figure 39, where random choices ought not to go! How was that set found? By picking out the one and only random set of choices that will lead there. But millions of other sets of random choices would lead to figure 41, so the chance of hitting upon just this one set by accident, rather than choosing it with hindsight, is negligibly small (though not zero).
32a Very nearly 2.0.
32b Very nearly 2.0.
32c Very nearly 2.0.
32d The ratios of numbers of atoms in adjacent energy levels is constant, and equal to 2.0.
32e The ratio of numbers of atoms decaying in successive equal time intervals is constant. The decay has an exponential form.
32f Exponential.
33a Yes, pretty well constant.
33b No. The ratio is close to 4.0.
33c See table 13.
Figure No. N q N/q 1+N/q Distribution ratio 42 900 900 1.0 2.0 ≈ 2.0 60 900 300 3.0 4.0 ≈ 4.0 65 1800 1200 1.5 2.5 ≈ 2.5 (both) 50 625 625 1.0 2.0 ≈ 2.0
Table 13: Comparison of predictions of distribution ratios with results from the computer simulation on film.
34 In figure 50 there are fewer atoms in all than in figure 42, and there are fewer atoms in each energy level. But in both there are twice as many atoms in each level as in the adjacent higher level.
They are the same in this sense: they have the same proportion of atoms to quanta.
35a No. It was told to move quanta at random.
35b No. They came about quite by chance. Another time the energies could be different.
35c Yes. It happens every time we try.
36a From the less sloping, or shallower distribution, to the more steeply sloping distribution.
36b From high temperature to low.
36c Their general shape no longer changes, apart from fluctuations around that shape. The two distributions continue to have the same shape, and the same steepness, as each other.
36d When the casserole's temperature is equal to that of the oven walls.
36e The lefthand half was hotter than the righthand half in figure 63. In figure 64, the two halves are equally hot.
36f Large ratios of numbers of atoms in adjacent energy levels go with low temperatures, and vice versa.
37a There is a steady small chance that one will get a cold. Because the school has some hundreds of students, there is usually just about the expected proportion of students absent.
37b No - at least we didn't tell it!
37c Very fair. In fact it sums it up nicely.