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water will fall through the air with uniform speed. At first, when it begins to fall, the weight of the drop will be greater than the friction; but as the speed of the drop increases, the friction will increase. The drop soon reaches that speed where the friction will be equal to its weight, and the resultant force becomes zero. It has been found possible to compute the size of small drops of water by measuring the rate at which they fall through the air and applying the complicated laws of viscosity.

It is worth while to explain why small drops fall at such low speeds; for example, the fine drops which constitute a cloud. The downward force acting on each drop is its weight, and the retarding force is the viscosity of the air. The weight is proportional to the volume of the drop, but the viscosity depends on the surface of the drop. Since the volume is proportional to the cube of the diameter and the surface to the square of the diameter, as the drop gets smaller its volume decreases much faster than its surface. Hence small drops have less weight as compared with the viscosity effect than do large ones. The result is that the retarding force on a very small drop becomes equal to its weight at low speeds. For the case where the diameter of a drop of water is one fiftieth of a millimeter, estimates show that it will fall at a uniform speed of 1.2 centimeters per second.

On the other hand, large raindrops must fall much faster in order for the retarding force to be equal to their weight. But there is a limit to the speed with which large drops of water can fall through the air. Whenever a drop exceeds a speed of about 8 meters (26 feet) per second, it is broken up into smaller drops, which cannot fall so fast.

Considerable attention has been given to the increase of fluid friction at high speeds. It is only at low speeds that the friction is proportional to the velocity. At higher speeds the friction varies approximately as the square of the speed, and at still higher speeds it increases more rapidly than that. As the speed of a train increases, the air friction increases very rapidly, until at high speeds it becomes greater than the other resistances, although these may also increase with the speed. It is this increase of friction with speed which makes it so expensive to run automobiles, trains, and ships at high speeds. It requires so much coal to drive a destroyer or a battle cruiser at its maximum speed that the vessel cannot carry enough to run at full speed for more than a comparatively few hours.

An increase in temperature lowers the viscosity of liquids but increases the viscosity of gases.

The laws of friction as applied to lubricated surfaces are also complicated. It appears, however, that when the surfaces are well lubricated, the friction is proportional to the relative velocities of the two surfaces.

111. Viscosity in solids. There is an internal friction in solids. The vibrations of a tuning-fork die out at a rate greater than can be explained by the rate at which energy is given to the air. A vibrating spring, or any vibrating solid, has internal friction which helps bring it to rest. The internal friction, or viscosity, in metals not only varies with different metals, but in the same metal it varies with the annealing. There is also internal friction in belts and cables used in driving machines.

112. Viscosity in another sense. The term viscous is often applied to solids which cannot resist the continued application of a force, although able to resist it for a short time. A piece of pitch struck by a hammer may fly into small pieces; yet if left alone long enough, under the continued action of its own weight, it will flow like a liquid. A piece of sealing-wax supported in a horizontal position by one end only will slowly bend under its own weight.

However, most metals show no signs of this sort of viscosity. The sharp outlines of the impressions on very old coins, and the preservation of old pieces of metallic art, show that this type of viscosity does not exist to any appreciable amount in these metals.

PROBLEMS

1. It required a horizontal force of 50 lb. to drag a 200-pound weight over a smooth floor at a uniform speed. Compute the coefficient of friction. 2. A force of 150 lb. is required to pull with uniform speed a sled which with its load weighs 2000 lb. Find the coefficient of friction.

3. A horizontal force of 40 lb. gives to a 320-pound weight a horizontal acceleration of 2 ft./sec.2 (a) What is the frictional force? (b) What is the coefficient of friction ?

4. If the coefficient of friction is 0.2 and the load is 1000 lb., how much work will be done in moving the load at constant speed for 200 ft.?

5. If the coefficient of starting friction between the driving-wheel of a locomotive and a rail is 0.3, what minimum weight must the driving-wheel carry in order that the pull exerted through this wheel on the train may be 1 ton?

6. A 3000-pound automobile will coast down a 5 per cent grade at a constant speed of 20 mi./hr. What is the retarding force on the car?

7. On a horizontal road a force of 200 lb. gives a 3200-pound car an acceleration of 1 ft./sec.2 What is the retarding force?

8. A mass which weighs 400 lb., starting from rest at the top of an incline 30 ft. high, acquires a speed of 20 ft./sec. at the bottom. How much energy was lost on account of friction?

9. A force of 200 lb. was necessary to slide a 300-pound box up an incline 12 ft. long and 4 ft. high. Find the force of the friction.

10. A shaft 1 in. in diameter has a total load of 250 lb. pushing it against its bearings. Compute the force necessary to overcome the friction if the coefficient of friction is 0.05 and if the force is applied to the rim of a pulley 18 in. in diameter, which is mounted on the shaft.

11. A wagon with a total load of 5000 lb. has wheels 3 ft. in diameter and axles 2 in. in diameter. If the coefficient of sliding friction at the axle is 0.15 and the coefficient of rolling friction on a certain hard road is 0.025, what will be the necessary horizontal force, applied at the axle, to keep the wagon moving at a constant speed?

12. What will be the force required to keep a 100,000-pound car moving on a level track if the coefficient of sliding friction at the axle is 0.05 and the coefficient of rolling friction is 0.002, and if the diameters of the wheel and axle are 30 in. and 2 in. respectively?

13. (a) Find the work done by a horse in hauling a sled weighing 1600 lb. for a mile along a level road if the coefficient of friction is 0.05. (b) If the horse travels 3 mi./hr., what power does he deliver?

14. The frictional force on the surface of a 1-inch shaft is 5 lb. How much power is expended in overcoming this friction if the shaft makes 300 revolutions per minute?

CHAPTER X

VIBRATORY MOTION

Vibratory motion, 113. Simple harmonic motion, 114. Another definition of simple harmonic motion, 115. Proof of the equality of the two definitions, 116. Period, amplitude, phase, 117. The fundamental equation of simple harmonic motion, 118. The simple pendulum, 119. The physical, or compound, pendulum, 120. The torsion pendulum; angular simple harmonic motion, 121. Vibrations in musical instruments, 122.

113. Vibratory motion. Ever since Galileo watched a swaying chandelier in a cathedral in Pisa and, partly by observation and partly by guesswork, found one of the laws of its motion, the vibrations of pendulums have been of practical and theoretical interest. The oscillations of a pendulum are only one case of what is called vibratory motion. In all musical instruments sound is produced by something which has a vibratory motion. There is some evidence to show that Pythagoras (500 or 600 в.с.) worked out some of the simpler laws of the vibrations of the strings of musical instruments.

The principles which we shall study apply not only to cases such as have been mentioned but also to wave-motions; for wavemotions, such as waves traveling over the surface of water, are a special type of vibratory motion. They apply also to light-waves and electrical waves, where the vibrations are not those of a material medium. The detailed study of alternating electric currents, the kind ordinarily used in lighting-circuits, is to a large extent an application of the principles of vibratory motion.

That type of vibratory motion which, all things considered, is the simplest is called simple harmonic motion. The use of the term harmonic arose on account of the fact that the first detailed study of this subject was made in connection with the study of musical vibrations.

114. Simple harmonic motion. If a mass which is suspended by a spiral spring is pulled down, and then released, it will vibrate up and down with simple harmonic motion. In this case it is not difficult to state the facts relating to the magnitude and direction of the resultant force acting on the mass.

1. When the mass is pulled down, there is an upward force acting; if the mass is pushed up, there is a downward force. There is always a "return" force tending to restore the mass to its initial position of rest, the position of equilibrium.

2. It is well known that when a spring is stretched or compressed, the reacting force is proportional to the change of length of the spring. It follows from this that the return force produced by the spring is directly proportional to the distance the mass is displaced from its position of equilibrium.

These facts are characteristic of simple harmonic motion. Hence we may define simple harmonic motion as follows: In simple harmonic motion there is at each instant a restoring force proportional to the distance the body is displaced from its equilibrium position.

Whenever it is known that the restoring force is proportional to the displacement, then it is known that the vibrations are simple harmonic vibrations. The laws of elasticity tell us that this is true in the case of the spiral spring, in the case of thin steel rods clamped at one end, and in many other cases involving elastic forces. In the case of a pendulum, or a boy in a swing, it is proved in section 119 that the restoring force is proportional to the displacement, provided the amplitude of motion is small.

Hence in these cases the motion is simple harmonic motion.

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M

N
FIG. 88

C

115. Another definition of simple har- Ad monic motion. A ball B is mounted on any suitable support AC, which can be rotated about an axis MN (Fig. 88). The type of rotator commonly used on the classroom table is satisfactory. When the ball is revolving around the axis MN, an eye situated in the plane in which the ball is moving will see only a vibratory motion. Or the rotating ball may be placed in a beam of light in such a position that its shadow

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