Mouse eating cat

All of the 13 mice in the figure are doomed to be eaten by the cat. But the cat wants to eat them in a certain order. The cat eats one mouse and then counts around the circle in the direction in which the mice are looking. When it gets to 13 it eats the next mouse and starts counting again. Which mouse must the cat eat first so that the white mouse is eaten last?

Source: Fun with Maths and Physics by Perlman 

Fun Physics: Centrifugal Force

Open an umbrella, put its end on the floor, spin it and drop a ball into it. The ball could be a balled piece of paper or handkerchief, or any other light and unbreakable thing. Something will happen you probably wouldn't expect. The umbrella does not accept the present and the thing will crawl up the edge and then flies off in a straight line. 

The force that threw the ball out in this experiment is generally called the "centrifugal force", although it would be more appropriate to dub it "inertia". Centrifugal force manifests itself when a body travels in a circle but this is nothing but an example of inertia which is the desire of a moving body to maintain its speed and direction. 

We come across centrifugal force more often than you might suspect. If you whirl a stone tied to a piece of string, you can feel the string become taut and seem to be about to break under the action of the centrifugal force. The ancient weapon for hurling stones, the sling, owes its existence to the force. Centrifugal force bursts a millstone, if it is spun too fast and is not sufficiently strong. If you are adroit enough, this force will help you to perform a trick with a glass from which the water doesn't escape, even though it is upside down. In order to do this you'll only have to swing the glass quickly above your head in a circle. Centrifugal force helps a circus bicyclist to do a "devil's loop". It is put to work. In the so-called centrifugal separators it churns cream; it extracts honey from honey-comb; it dries washing by extracting water in centrifugal driers, etc., etc. 


When a tram travels in a circular path, e.g. as it turns at a crossing, the passengers feel directly the centrifugal force that pushes them in the direction of the outer wall of the carriage. If the speed is sufficiently large, the carriage could be overturned by the force if the outer rail wasn't laid a bit higher than the inner one: which is why a tram is slightly inclined inwards when it turns. It sounds rather unusual but an inclined tram is more stable than an upright one! 

But this is quite the case, though. A small experiment will help explain this to you. Bend a cardboard sheet to form a wide funnel, or better still take a conical bowl if available. The conical shield (glass or metallic) of an electrical lamp would be suitable for our purposes. Roll a coin (small metal disk, or ring) around the edge of any of these objects. It will travel in a circle bending in noticeably on its way. As the coin slows down, it will travel in ever decreasing circles approaching the centre of the funnel. But by slightly shaking the funnel the coin can easily be make roll faster and then it will move away from the centre describing increasingly larger circles. If you overdo it a bit, the coin will roll out. 

For cycling races in a velodrome special circular tracks are made and you can see that these tracks, especially where they turn abruptly have a noticeable slope into the centre. A cyclist rides along them in an inclined position  like the coin in the funnel) and not only does he not turn over but he acquires special stability. Circus cyclists used to amaze the public by racing along a steep deck. Now you can understand that there is nothing special about it. On the contrary, it would be a hard job for a cyclist to travel along a horizontal track. For the same reason a rider and his horse lean inwards on a sharp turn. 

Let's pass on from small to large-scale phenomena. The Earth, on which we live, rotates and so centrifugal force should manifest itself. But where and how? By making all the things on its surface lighter. The closer something is to the Equator, the larger the circle in which it moves and hence it rotates faster, thereby losing more of its weight. If a 1-kg mass were to be brought from one of the poles to the Equator and reweighed using a spring balance, the loss in weight would amount to 5 grammes. That, of course, is not very much of a difference, but the heavier a thing, the larger the difference. A locomotive that has come from Stockholm to Rome loses 60 kg, the weight of an adult. A battle ship of 20,000-tonne displacement that has come from the White Sea to the Black Sea will have lost as much as 80 tonnes, the weight of a locomotive! 

Watch the "Can we make it to Mars?" again video between time 9.30 to 11.15



Why does it happen? Because as the globe rotates, it tries to throw everything off its surface just like the umbrella in our earlier experiment. It would succeed were it not for the terrestrial attraction that pulls everything back to the Earth's surface. We call this attraction "gravity". The rotation cannot throw things off the Earth's surface, but it can make them lighter. 

The faster the rotation, the more noticeable the reduction in weight. Scientists have calculated that if the Earth rotated 17 times faster, things at the Equator would lose their weight completely to become weightless. And if it rotated yet quicker, making, say, one turn every hour, then the weight lessness would extend to the lands and seas farther away from the Equator. 

Just imagine things losing their weight. It would mean there would be nothing you could not lift, you would be able to lift locomotives, boulders, cannons and warships as easily as you could a feather. And should you drop them-no danger, they could hurt nobody since they wouldn't fall down at all, but would float about in mid-air just where you'd let go of them. If, sitting in the cabin of an airship, you wanted to throw something overboard, it wouldn't drop, but would stay in the air. What a wonder world it would be. So you could jump as high as you've never dreamed, higher than sky-scrapers or the mountains. But remember, it would be easy to jump up but  difficult to return back to ground. Weightless, you'd never come back on your own. 

There would also be other inconveniences in such a world. You've probably realized yourself that everything, whatever its size, would, if not fixed, rise up due to the slightest motion of air and float about. People, animals, cars, carts, ships-everything would move about in the air disorderly, breaking, maiming and destroying. That is what would occur if the Earth rotated  significantly faster.

Racing against time

Could one leave Vladivostok by air at 8 a.m. and land in Moscow at 8 a.m. on the same day? I'm not talking through my hat. We can really do that. 

The answer lies in the 9-hour difference in Vladivostok and Moscow zonal times(just like India and US are in different zonal times). If our plane covers the distance between the two cities in these 9 hours, it will land in Moscow at the very same time at which it took off from Vladivostok. Considering that the distance is roughly 9,000 kilometers, we must fly at a speed of 9,000:9 = 1,000 km/hour, which is quite possible today. 

Outrace the Sun

To "outrace the Sun" (or rather the earth) in Arctic latitudes, one can go much more slowly. Above Novaya Zemlya, on the 77th parallel, a plane doing about 450 km/hour. would cover as much as a definite point on the surface of the globe would cover in an identical space of time in the process of the earth's axial rotation. 

If you were flying in such a plane you would see the sun suspended in immobility. It would never set, provided, of course, that your plane was moving in the proper direction. 

Outrace the Moon

It is still easier to "outrace the Moon" in its revolution around the earth. It takes the moon 29 times longer to spin round the earth than it takes the earth to complete one rotation. So any ordinary steamer boat making 15-18 knots (1 knot = 1.852 km/hour) could "outrace the Moon" oven in the moderate latitudes. 

Mark Twain (a writer) mentions this in his Innocents Abroad. When sailing across the Atlantic, from New York to the Azores "... we had balmy summer weather, and nights that were even finer than the days. We had the phenomenon of a full moon located just in the same spot in the heavens at the same hour every night. The reason for this singular conduct on the part of the moon did not occur to us at first, but it did afterward when we reflected that we were gaining about twenty minutes every day, because we were going east (moon rotates from west to east around earth) so fast we gained just enough every day to keep along with the moon. "

In Six Rows - Riddle

You may have heard the funny story that nine horses have been put into 10 boxes, one in each. The problem that is now posed is formally similar to this famous joke, but it has a real solution *. You must arrange 24 people in six rows with five in each.

Fun Physics: Standing an egg

A schoolboy once wrote in a composition:  "Christopher Columbus was a great man because he discovered America and stood an egg upright." This young scholar had thought both deeds equally amazing. On the contrary, the American humorist Mark Twain saw nothing special about Columbus discovering America: "It would have been strange if he hadn't found it there." 

The other feat the great navigator had performed is not really all that marvelous. Do you know how Columbus stood an egg upright? He simply pressed it down onto a table crushing the bottom of the shell. He had, of course, changed the shape of the egg. But how can one possibly stand an egg on end without changing its shape, the navigator didn't know. 

Meanwhile it is easier by far than discovering America or even one tiny island. I'll show you three methods: one for boiled eggs, one for raw eggs, and one for both. 

A boiled egg can be stood upright simply by spinning it with your fingers or between your palms like a top. The egg will remain upright as long as it spins. After two or three trials the experiment should come out well.

This won't work if you try to stand a raw egg  upright, you may have noticed that raw eggs spin poorly. This, by the way, is used to distinguish a hard-boiled egg from a raw one without breaking the shell. The liquid contents of a raw egg is not carried along by the spinning as fast as the shell and, therefore, sort of
damps the speed down. We have to look for another way of standing eggs and one does exist. You have to shake an egg intensely several times. This breaks down the soft envelope containing the yolk with the result that the yolk spreads out inside the egg. If you then stand the egg on its blunt end and keep it this way for a while, then the yolk, which is heavier than the white, will pour down to the bottom of the egg and concentrate there. This will bring the centre of mass of the egg down making it more stable than before.

Finally, there is a third way of putting an egg  upright. If an egg is placed, say, on the top of a corked bottle and another cork with two forks stuck into it is placed on the top as shown in Fig. 17, the whole system (as a physicist would put it) is fairly stable and remains in equilibrium even if the bottle is slightly
inclined. But why don't the egg and cork fall down? For the same reason that a pencil placed upright on a finger doesn't fall off when a bent penknife is stuck into it as shown. A scientist would explain: "The centre of mass of the system lies below the support." This means that the point at which the weight of the system is applied lies below the place at which it is supported.

A Squirrel in the Glade

I had quite a bit of fun playing hide-and-seek with a squirrel," he said. "You know that little round glade with a lone birch in the centre? It was on this tree that a squirrel was hiding from me. As I emerged from a thicket, I saw its snout and two bright little eyes peeping from behind the trunk. I wanted to see the little animal, so I started circling round along the edge of the glade, mindful of keeping the distance in order not to scare it. I did four rounds, but the little cheat kept backing away from me, eyeing me suspiciously from behind the tree. Try as I did, I just could not see its back."
"But you have just said yourself that you circled round the tree four times," one of the listeners interjected.
"Round the tree, yes, but not round the squirrel."
"But the squirrel was on the tree, wasn't it?"
"So it was."
"Well, that means you circled round the squirrel too."
"Call that circling round the squirrel when I didn't see its back"

"What has its back to do with the whole thing? The squirrel was on the tree in the centre of the glade and you circled round the tree. In other words, you circled round the squirrel."
"Oh no, I didn't. Let us assume that I'm circling round you and you keep turning, showing me just your face. Call that circling round you?"
"Of course, what else can you call it?"
"You mean I'm circling round you though I'm never behind you and never see your back?"
"Forget the back! You're circling round me and that's what counts. What has the back to do with it?"
"Wait. Tell me, what's circling round anything? The way 1 understand it, it's moving in such a manner so as to see the object I'm moving around from all sides. Am I right, professor?" He turned to an old man at our table.
"Your whole argument is essentially one about a word," the professor replied. "What you should do first is agree on the definition of 'circling'. How do you understand the words 'circle round an object'? There are two ways of understanding that. First, it's moving round an object that is in the centre of a circle. Secondly, it's moving round an object in such a way as to see all its sides. If you insist on the first meaning, then you walked round the squirrel four times. If it's the second that you hold to, then you did not walk round it at all. There's really no ground for an argument here, that is, if you two speak the same language and understand words in the same way."
"All right, I grant there are two meanings. But which is the correct one?"
"That's not the way to put the question. You can agree about anything. The question is, which of the two meanings is the more generally accepted? In my opinion, it's the first, and here's why. The sun, as you know, does a complete revolution in a little more than 25 days....
"Does the sun revolve?" Part One
"Of course, it does, like the earth around its axis. Just imagine, for instance, that it would take not 25 days, but 365 1/4 days, i. e. a whole year, to do so. If this were the
case, the earth would see only one side of the sun, that is, only its 'face'. And yet, can anyone claim that the earth does not revolve round the sun?"
"Yes, now it's clear that I circled round the squirrel after all."

Dictionary
Glade - Jungle
Birch - a kind of tree