WHEN WE MOVE ROUND THE SUN FASTER... Physics can be entertainment

WHEN WE MOVE ROUND THE SUN FASTER

Paris newspapers once carried an ad offering a cheap and pleasant

way of travelling for the price of 25 centimes. Several simpletons

mailed this sum. Each received a letter of the following

content:

"Sir, rest at peace in bed and remember that the earth turns. At the

49th parallel that of Paris you travel more than 25,000 km a day.

Should you want a nice view, draw your curtain aside and admire the

starry sky."

The man who sent these letters was found and tried for fraud. The

story goes that after quietly listening to the verdict and paying the

fine demanded, the culprit struck a theatrical pose and solemnly declared,

repeating Galileo's famous words: "It turns. 1 '

He was right, to some extent, after all, every inhabitant of the

globe "travels" not only as the earth rotates. He is transported with

still greater speed as the earth revolves around the sun. Every second this

planet of ours, with us and everything else on it, moves 30 km in space,

turning meanwhile on its axis. And thereby hangs a question not devoid

of interest: When do we move around the sun faster? In the daytime

or at night?

A bit of a puzzler, isn't it? After all, it's always day on one side of

the earth and night on the other. But don't dismiss my question as

senseless. Note that I'm asking you not when the earth itself moves

faster, but when we, who live on the earth, move faster in the heavens.

And that is another pair of shoes.

In the solar system we make two motions; we revolve around the

sun and simultaneously turn on the earth's axis. The two motions

add , but with different results, depending whether we are on the daylit

side or on the nightbound one.

Fig. 6 shows you that at midnight the speed of rotation is added to

that of the earth's translation, while at noon it is, on the contrary,

subtracted from the latter. Consequently, at midnight we move faster

in the solar system than at noon. Since any point on the equator travels

about half a kilometre a second, the difference there between midnight

and midday speeds comes to as much as a whole kilometre a second.

21

Midday

Midnighi

Fig. 6. On the dark side we move around the sun faster

than on the sunlit side

Any of you who are good at geometry will easily reckon that for

Leningrad, which is on the 60th parallel, this difference is only half as

much. At 12 p.m. Leningraders travel in the solar system half a

kilometre more a second than they would do at 12 a.m.

THE CART-WHEEL RIDDLE

Attach a strip of coloured paper to the side of the rim of a cart-wheel

or bicycle tire, and watch to see what happens when the cart, or bicycle,

moves. If you are observant enough, you will see that near the ground

the strip of paper appears rather distinctly, while on top it flashes by

so rapidly that you can hardly spot it.

Doesn't it seem that the top of the wheel is moving faster than the

bottom? And when you look at the upper and lower spokes of the moving

wheel of a carriage, wouldn't you think the same? Indeed, the upper

spokes seem to merge into one solid body, whereas the lower spokes

can be made out quite distinctly.

22

Incredibly enough, the top of the rolling-wheel does really move faster

than the bottom. And, though seemingly unbelievable, the explanation

is a pretty simple one. Every point on the rolling wheel makes two

motions simultaneously one about the axle and the other forward

together with the axle. It's the same as with the earth itself. The two

motions add, but with different results for the top and bottom of the

wheel. At the top the wheel's motion of rotation is added to its motion

of translation, since both are in the same direction. At the bot

torn rotation is made in the reverse direction and, consequently, must

be subtracted from translation. That is why the stationary observer

sees the top of the wheel moving faster than the bottom.

A simple experiment which can be done at convenience proves this

point. Drive a stick into the ground next to the wheel of a stationary

vehicle opposite the axle.Then take a piece of coal or chalk and make two

marks on the rim of the wheel at the very top and at the very bottom.

Your marks should be right opposite the stick. Now push the vehicle

a bit to the right (Fig. 7), so that the axle moves some 20 to 30 cm away

from the stick. Look to see how the marks have shifted. You will find

that the upper mark A has shifted much further away than the lower

one B which is almost where it was before.

Fig. 7. A comparison between the distances away from

the stick of points A and B on a rolling wheel (right) shows

that the wheel's upper segment moves faster than its lower

part

THE WHEEL'S SLOWEST PART

As we have seen, not all parts of a rolling cart-wheel move with the

same speed. Which part is slowest? That which touches the ground.

Strictly speaking, at the moment of contact, this part is absolutely

stationary. This refers only to a rolling wheel. For the one that spins

round a fixed axis, this is not so. In the case of a flywheel, for instance,

all its parts move with the same speed.

BRAIN-TEASER

Here is another, just as ticklish, problem. Could a train going from

Leningrad to Moscow have any points which, in relation to the railroad

track, would be moving in the opposite direction? It could, we find.

All the train wheels have such points every moment. They are at the

bottom of the protruding rim of the wheel (the bead). When the train

goes forward, these points move backward. The following experiment,

which you can easily do yourself, will show you how this happens.

Attach a match to a coin with some plasticine so that the match pro*

trades in the plane of the radius, as shown in Fig. 8. Set the coin together

with the match in a vertical position on the edge of a flat ruler and

hold it with your thumb at its point of contact C. Then roll it to and

fro. You will see that points F, E and D of the jutting part of the match

Fig. 8. When the coin is rolled

leftwards, points Ft E and

D of the jutting part of the

match move backwards

Fig. 9. When the train wheel

rolls leftwards the lower part

of its rim rolls the other way

/ig. 10. Top: the curve (a cycloid) described by every

point on the rim of a rolling cart-wheel. Bottom: the curve

described by every point on the rim of a train wheel

move not forwards but backwards. The further point D the end of the

match is from the edge of the coin, the more noticeable backward

motion is (point D shifts to D').

The points on the bead of the train wheel move similarly. So when

I tell you now that there are points in a train that move not forward

but backward, this should no longer surprise you. True, this backward

motion lasts only the negligible fraction of a second. Still there is,

despite all our habitual notions, a backward motion in a moving train.

Figs. 9 and 10 provide the explanation.

WHERE DID THE YACHT CAST OFF?

A rowboat is crossing a lake. Arrow a in Fig. 11 is its velocity vector.

A yacht is cutting across its course; arrow b is its velocity vector.

Where did the yacht cast off? You would naturally point at once to

point M. But you would get a different reply from the people in the

dinghy. Why?

They don't see the yacht moving at right angles to their own course,

because they don't realise that they are moving themselves. They think

25

Fig. 11. The yacht is cutting across the rowboat's course. Arrows a and b designate

the velocities. What will the people in the dinghy see?

they're stationary, while everything around is moving with their own

speed but in the opposite direction. From their point of view the yacht

is moving not only in the direction of the arrow b but also in the direction

of the dotted line a opposite to their own direction (Fig. 12).

The two motions of the yacht the real one and the seeming one are

resolved according to the rule of the parallelogram. The result is that

the people in the rowboat think the yacht to be moving along the

diagonal of the parallelogram 06; that is also why they think the yacht

cast off not at point M, but at point /V, way in front of the rowboat

(Fig. 12).

Travelling together with the earth in its orbital path, we also plot

the position of the stars wrongly just as the people in the dinghy did

when asked where the yacht cast off from. We see the stars displaced

slightly forward in the direction of the earth's orbital motion. Of course,

the earth's speed is negligible compared with that of light (10,000

Fig. 12. The people in the dinghy think the yacht to be coming towards them

slantwise from point N

times less) and, consequently, this stellar displacement, known as

aberration of light, is insignificant. However, we can detect it with

the aid of astronomical instruments.

Did you like the yacht problem? Then answer another two questions

related to the same problem. Firstly, give the direction in which the

yachtsmen think the dinghy is moving. Secondly, say where the yachts*

men think the dinghy is heading. To answer, you must construct a parallelogram

of velocities on the vector a (Fig. 12), whose diagonal will

indicate that from the yachtsmen's point of view the dinghy seems to

be moving slantwise, as if heading for the shore.