At B the centrifugal force is less than at A, but there the direct attraction of the moon tends to lift the water. The result is that there is a reduction of the weight of water at both A and B, and hence a tendency to produce tides at those places.*

MX

In addition to the motion already described, the earth has a rotation around the axis MN with a period of twenty-four hours. This is the motion which gives us day and night. The earth spinning on the axis MN carries through the regions marked A and B all parts of the surface of the earth which are not near the poles. The surface of the earth is thus forced through the tides; or, as they appear to those who are moving with the earth, the tides travel around the earth as extremely long waves. The continents and other masses of land A E interfere with the free motion of these waves, so that the actual direction and magnitude of the wave-motion become very complicated.

NO

B

m

FIG. 104

The sun produces an effect, but a much smaller one than that produced by the moon; in fact, an effect so small that it is usually masked by the tide due to the moon. At both new and full moon the tide-producing forces of the moon and the sun act together, producing larger tides, which are called spring tides. This condition occurs twice a month.

Since the cause of the tidal flow is the rotation of the earth about the axis MN, the source of energy in the tides is the kinetic energy of rotation of the earth. Friction in the ebb and flow of the tides should tend to slow up the earth's rotation.

PROBLEMS

1. What would a body which weighed 100 lb. on the earth's surface weigh when placed at a distance equal to the earth's radius above the surface?

2. Compute the force of attraction in dynes between two spheres of 5 kg. each which are placed with their centers 20 cm. apart.

3. What is the ratio of the weight of a kilogram on the surface of the earth to what it would be if raised 10 km. from the surface? (Assume the radius of the earth to be 6.3 × 108 cm.)

* The attraction the moon exerts on the water at A decreases the effect of the rotation about the axis OX, but the necessary centripetal force is much greater than the attraction of the moon; hence water has less weight at A. Computations show that the losses of weight of unit mass of water at A and B are approximately equal.

4. Find the ratio of the weight of a body in Madison, Wisconsin, to what it would be in New Orleans (see section 142).

What would be the acceleration of a freely falling body if the body is placed 1000 mi. above the earth's surface? (Assume the diameter of the earth to be 7920 mi. and g, at the surface, to be 32.2 ft./sec.2)

How many pounds will a man who weighs 150 lb. on the earth weigh on the surface of the moon? (The mass of the moon is 0.0128 of that of the earth, and the radius of the moon is 0.273 of the earth's radius.)

A lead ball, the mass of which is equal to 10 kg., is suspended from the arm of a balance. (a) What will be the attractive force on it when a 200kilogram ball is placed directly under it, the distance between centers being 25 cm.? (b) What will be the apparent increase in the mass of the suspended ball?

CHAPTER XIII

FLUIDS IN MOTION

Loss of pressure due to fluid friction, 145. Change in pressure produced when speed of liquid is changed, 146. Explanation of decrease in pressure with increased speeds, 147. Illustrations of decrease of pressure with increase in speed, 148. Bernoulli's theorem, 149. Proof of Bernoulli's theorem, 150. Summary, 151. Energy of a liquid, 152. Torricelli's theorem, 153. Transference of hydraulic energy of water to machines, 154. Water wheels and turbines, 155. Hydraulic ram, 156. Pitot tube, 157.

Any understanding of the simple properties of fluids that does not include facts regarding fluids in motion is incomplete. Some of the facts which are true for liquids at rest are not always true for moving liquids. An example in point is the theorem which states that at all points on the same level in connected vessels the pressure is the same (sect. 23).

The use of water power has become a matter of great economic importance; hence it is of considerable interest to know something about the principles of moving liquids.

145. Loss of pressure due to fluid friction. The most noticeable effect of fluid friction is in the loss of pressure in pipe lines. Force must be used to overcome the retarding forces of friction. Hence in any pipe line where water or any other liquid is flowing, there is a continuous decrease in pressure as the distance from the source of supply increases.

In pipes of the same size the loss of pressure on account of friction is proportional to the length of pipe and approximately proportional to the velocity of the fluid. The loss of pressure in large pipes is less than in smaller ones, provided the velocity of flow is the same in the two.

146. Change in pressure produced when speed of liquid is changed. Fig. 105 shows diagrammatically an experiment in which the water is flowing through a constriction in a pipe. The liquid has a greater speed at B than at either A or C. The height of the water in the vertical tubes shows that the pressure in the pipe is greater at A and C than at B. The pressure at C will be somewhat less than that at A on account of the fluid friction. But the large decrease in pressure from A to

there is a decrease in pressure wherever the speed is increased, and an increase in pressure wherever the speed is decreased. 147. Explanation of decrease in pressure with increased speeds. The pressure of the liquid at B of Fig. 105 is transmitted through the liquid at A from the pump or source of pressure. But the liquid between A and B is being accelerated. No body which is being accelerated transmits all the force or pressure which is applied to it. Take the simple case of a book lying in one's hand. The earth exerts a downward force on the book. This is transmitted to the hand, producing an equal downward force on it. But if the hand and the book are accelerated downward, not all the force due to the weight of the book is transmitted to the hand: the pressure on the hand is decreased because part of the force is used in accelerating the book. So it is that the pressure transmitted to B is less than at A, because part of it is used in accelerating the water between A and B. In the space between B and C there is a negative acceleration, and the pressure is greater at C than at B.

148. Illustrations of decrease of pressure with increase in speed. Let A be a point in the water of a reservoir, and B a point in the outlet pipe (Fig. 106). Although the points A and B lie on the same level, the pressures will not be the same at these points, for at B the speed of the liquid is much greater than at A. As explained in the last section, this is due to the fact that the water between A and B is being accelerated. The observed decrease in pressure in the water pipes in a house when water is flowing from one of the faucets is due partly to this effect.

Two light balls (for example, hollow celluloid balls) are suspended by strings so that they hang two or three inches apart. If one blows in between the balls, they will move together at right angles to the direction of the blast. In Fig. 107 the direction of the motion of the blast of air is shown by the lines AA'. Between the balls the air has considerable speed, while beyond, at A', where the blast spreads out, the speed is much less. It was shown in connection with Fig. 105 that the pressure at B must be less than at C. For a similar reason the pressure between the balls must be less than at A'. At A', or a little beyond, where the air comes to

FIG. 106

FIG. 107

F

rest, the pressure must be the normal pressure of the air. Hence the pressure between the balls is less than atmospheric pressure, and they are pushed together by the air pressure on the outside. A similar thing happens when two boats are lying side by side in a current of water. It is a well-established fact that observed phenomena depend only on relative motion and not on absolute motion. Since there are forces pushing the boats together when they are in moving water, it follows that if they are moving side by side in stationary water there will also be forces pushing them together. For the relative motion of the boats and the water is the same in the two cases. Serious collisions have been caused when one boat in trying to pass another has come close enough to be affected by the apparent attraction. When airplanes began flying in squadrons, several wrecks occurred which were due probably to this cause.