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When we say then that a force applied to a particle is a certain multiple of another force, we mean that the first may be supposed to be composed of a certain number of forces equal to the second and acting in the same direction. In this way forces become measurable quantities, which can be expressed by numbers, like all other quantities, by referring them to a unit of their own kind. They may also be represented by straight lines proportional in length to these numbers, drawn from the point at which they act and in the directions in which they act.

9. Experience teaches us that if a body be let free from the hand, it will fall downwards in a certain direction; however frequently the experiment be made, the result is the same, the body strikes the same spot on the ground in each trial, provided the place from which it is dropped remain the same.

The cause of this undeviating effect is assumed to be an affinity which all bodies have for the earth, and is termed the force of attraction. If the body be prevented from falling by the interposition of a table or of the hand, the body exerts a pressure on the table or hand. Weight is the name given to the pressure which the attraction of the earth causes a body to exert on another with which it is in contact.

10. A solid body is conceived to be an aggregation of material particles which are held together by their mutual affinities. This appears to be a safe hypothesis, since experiments shew that any body is divisible into successively smaller and smaller portions without limit, if sufficient force be exerted to overcome the mutual action of the parts of the body.

11. A rigid body is one in which the particles retain invariable positions with respect to each other. No body in nature is perfectly rigid; every body yields more or less to the forces which act on it. If, then, in any case this compressibility is of a sensible magnitude, we shall suppose that the body has assumed its figure of equilibrium, and then consider the points of application of the forces as a system of invariable form. By body, hereafter, we mean rigid body.

12. When a force acts upon a body its effect will be unchanged in whatever point of its direction we suppose it applied, provided this point be either one of the points of the body or be invariably connected with it.

For suppose a body to be kept in equilibrium by a system of forces, one of which is the force P applied at the point A. Take any point B which lies on the direction of this force, and suppose B so connected with A that the distance AB is unchangeable. Then, if at B we introduce two forces, P and P', equal in magnitude and acting in opposite directions along the line AB, it is evident that no change is made in the effect of the force P at A. Also it is evident that P at A and P' at B will neutralise each other, and may therefore be removed without disturbing the equilibrium of the body. Hence, there remains the force P at B producing the same effect as when it acted at A.

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13. The principle established in the preceding article is known by the name of the transmissibility of a force to any point in its line of action. It is frequently assumed as an axiom, or as an experimental fact. When we find it useful to change the point of application of a force, we shall for shortness not always state that the new point is invariably connected with the old point, but this must be always understood.

CHAPTER II.

THE COMPOSITION AND EQUILIBRIUM OF FORCES ACTING

UPON A PARTICLE.

14. WHEN a particle is acted on by forces which do not maintain equilibrium, it will begin to move in some determinate direction. It is clear then that a single force may be found of such a magnitude, that if it acted in the direction opposite to that in which the motion would take place, this force would prevent the motion, and consequently would be in equilibrium with the other forces which act upon the particle. If then we were to remove the original forces and replace them by a single force, equal in magnitude to that described above, but acting in an opposite direction, the particle would still remain at rest. This force, which is equivalent in its effect to the combined effect of the original forces, is called their resultant, and the original forces are called the components of the resultant.

It will be necessary, then, to begin by deducing rules for the composition of forces ; that is, for finding their resultant force. After we have determined these, it will be an easy matter to deduce the analytical relations which forces must satisfy when in equilibrium.

15. PROP. To find the resultant of a given number of forces acting upon a particle in the same straight line ; and to find the condition which they must satisfy, that they may be in equilibrium.

When two or more forces act on a particle in the same direction, it is evident that the resulting force is equal to their sum, and acts in the same direction.

When two forces act in different directions, but in the same straight line, on a particle, it is equally clear that their resultant is equal to their difference, and acts in the direction of the greater component.

When several forces act in different directions, but in the same straight line, on a particle, the resultant of the forces acting in one direction equals the sum of these forces, and acts in the same direction: and so of the forces acting in the opposite direction. The resultant, therefore, of all the forces equals the difference of these sums, and acts in the direction of the greater.

If the forces acting in one direction are reckoned positive, and those in the opposite direction negative, then their resultant equals their algebraical sum; its sign determining the direction in which it acts.

In order that the forces may be in equilibrium, their resultant, and therefore their algebraical sum, must equal

zero.

16. There is another case in which we can easily determine the magnitude and direction of the resultant.

Let AB, AC, AD, be the directions of three equal forces acting on the particle A; suppose these forces all in the same plane and the three angles BAC, CAD, DAB each equal to 120°; the particle will remain at rest, for there is no reason why it should move in one direction rather than

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