each balance is kept constantly stretched, say to the 4-oz. point, so that they exert equal forces; then, if the masses after starting together keep pace with each other, they are acquiring velocity at the same rate and consequently are equal.

Of course such an experiment serves chiefly to illustrate what is meant by saying that equal masses have equal inertias, for it would be impossible to directly compare masses in this way with any degree of accuracy.

The actual comparison of masses is accomplished with great accuracy by weighing; for it is found that masses which have equal inertias have also equal weights, provided they are weighed in a vacuum at the same point on the earth.

(§38.)

33. Measure of Force.-A force may be measured in three ways:

1. By the weight that it can support. This is the gravitation method.

2. By its power to strain an elastic body, as in the ordinary spring balance.

3. By its power to give motion to a mass. This is the dynamical method.

The first method is very convenient and forms the basis of most measurements of force in engineering and ordinary life, but it has the disadvantage that the force required to support a pound weight varies from place to place on the earth.

The second method is convenient for comparing forces, but the elastic properties of one substance differ from those of another, besides being dependent on temperature and physical condition, so that a standard force could not be preserved or accurately defined by this method.

The third method is difficult to apply except indirectly, but furnishes a unit of force which depends only on the inertia of matter and is therefore absolutely invariable and well suited to be a standard force.

Forces are therefore defined according to the third method.

34. Equal Forces.-Two forces are said to be equal when the velocity of a given mass is increased at the same rate per second by one force as by the other. When two such forces act in opposite directions on a given mass they neutralize or balance each other so far as any effect on the motion of the mass is con

cerned. Thus when a cord is stretched horizontally between two springs, the forces exerted by the springs are equal and opposite so long as the cord remains at rest or moves with uniform velocity.

35. Force and Motion. It may be taken as a fundamental assumption of mechanics, which is in harmony with all our experience, that the effect of a force in changing the motion of a mass is not in any way affected by the state of rest or motion of the mass which is acted upon.

For instance, while a force is acting on a mass and increasing its velocity, suppose a second and equal force to act in the same direction upon the same mass. The second force being equal to the first will produce just as great an increase in velocity per second as is being produced by the first; and since both effects take place simultaneously and without interference, the total change in velocity will be twice that which would have been produced by the original force. It follows that the change in velocity per second when a force acts on a body is proportional to the amount of the force.

And the change in velocity of a body when acted on by a force is also proportional to the length of time during which the force acts, for suppose a mass has acquired velocity by a force acting upon it for one second, if the force now acts for another second it will increase the velocity of the mass as much more in the same direction since the effect of a force is in no way conditioned by the state of rest or motion of the body upon which it acts.

36. Impulse. The change in velocity which a given mass experiences is proportioned therefore both to the amount of the force and to the time during which it acts. A large force acting for a short time may produce the same change in the velocity of a mass as a small force acting for a longer time.

A billiard ball may be made to roll as fast by pushing it as by striking it with the cue; the force in the second case is very much greater than in the first, but is exerted during an exceedingly short time; the impulse in both cases must be the same.

The product of the amount of a force by the time during which it acts is called the impulse.

37. Force and Motion.-Again, suppose two equal masses

moving side by side are acted on by equal forces in the same direction, they will both gain in velocity equally and will accordingly continue to move side by side, and their motion will evidently not be affected in any way if the two masses are connected forming a single large mass.

From this consideration we see that if a force gives a certain acceleration to a given mass, then twice the force will be required to give the same acceleration to a mass twice as large, etc. Or, in order that different masses may all have the same change in velocity per second, the forces acting on them must be proportional to the masses.

But if the mass is doubled without any corresponding change in the force which acts upon it, the gain in velocity will only be half as great as before, for the motion in that case will be the same as if the original mass were acted on by half the original force.

38. Mass Proportional to Weight.-An interesting illustration of the case just discussed is found in falling bodies. Galileo concluded from his experiments that all kinds and sizes of bodies when dropped to the earth at the same place gained velocity at exactly the same rate, except for air resistance. But the force causing a body to fall is its weight. It follows, therefore, from the previous paragraph that the weights of bodies must be proportional to their masses, and for this reason masses may be compared by weighing them.

39. Momentum.-A given impulse may produce a great change in the velocity of a small mass or a proportionally smaller change in the velocity of greater mass; therefore, to measure the effect of an impulse, a quantity is employed which is proportional both to the mass and velocity of the moving body; this is called its momentum.

The momentum of a body is the product of the amount of its mass by the amount of its velocity, and is a directed or vector quantity.

40. Laws of Motion.-The relations between forces, masses and motion, were first clearly enunciated in the form of three laws of motion, by Sir Isaac Newton in his celebrated Principia, published in 1686.

First Law. Every body continues in its state of rest or of

moving with constant velocity in a straight line unless acted upon by some external force.

Second Law.-Change of momentum is proportional to the force and to the time during which it acts, and is in the same direction as the force.

Third Law. To every action there is an equal and opposite reaction.

41. Discussion of the First Law of Motion.-The first law asserts that force is not required to keep a body in motion, but simply to change its state of motion. After a railroad train has attained a constant speed the entire force of the locomotive is spent in overcoming the various resistances that oppose the motion, such as friction of wheels and bearings and air resistance. But for these the train would maintain its speed without aid from the locomotive.

Since we measure equal times by the equal angles through which the earth has moved, the law that freely moving bodies move through equal distances in equal times may seem simply a consequence of the mode of defining equal times and without any physical significance. But the statement of the law really asserts the physical fact that in case of any two bodies whatever, unacted on by external forces, while one body moves through successive equal distances, the distances traversed simultaneously by the other body are also equal among themselves.

That this is true whatever the nature of the bodies concerned is a fact of nature that rests on experience, and cannot be regarded as known a priori.

42. Discussion of Second Law. This law may be also expressed in the formula

mv-mu x Ft

where F is a force acting on a mass m for a time t, and u is the velocity at the beginning of the time interval t, while v is its velocity at the end of that time. Thus mv is the momentum after the force has acted, while mu is the original momentum of the mass. The gain in momentum is therefore my and according to the law this is proportional to the force F and to the time t jointly, or to their product Ft.

ти,

where k is a constant, the value of which depends on the particular units which are employed in measuring the various quantities concerned.

In the above equations F represents the average value of the force during the time t in which the velocity of the mass has

v-v

t

changed from u to v; but when t is exceedingly short approaches as its limit the actual rate of acceleration at the given instant, while F is the corresponding force at that same instant, and we may write,

that is, the acceleration of a body is proportional to the force acting upon it and inversely proportional to its mass.

This may be called the fundamental formula of dynamics as it is a direct expression of the second law of motion, is absolutely general, and enables us to determine the forces acting in any case where the mass and motion of a body are known, since the acceleration is determined from the motion.

Thus it follows that if the force acting on a mass is constant the mass moves with constant acceleration, while if the force varies the acceleration varies in the same proportion.

43. Stress. When a weight is supported by a uniform cord, every part of the cord is stretched, and if the weight of the cord itself is so small that it may be neglected, the stretch of every inch of it is the same whether it is near the upper or lower end and whatever may be the total length. The section A B (Fig. 9) is pulled up by the cord above A and is pulled down by the cord below B and is therefore stretched until the contractile force of its own elasticity balances the external stretching force. In this way every portion of the cord is subject from without to an external stretching force, and it exerts in opposition to this an internal contractile force and is said to be in a state of stress. In this case the stress is called tension, and each portion is subject to forces which tend to elongate it. When a weight