Tarasov Calculus test

September 9, 2020

First Book: On the General Laws of Equilibrium and Motion.

It is my intention to give in this book the general principles of the equilibrium and motion of bodies, and to solve those problems of mechanics which are indispensable in the theory of the system of the world.

Chapter I: On the Equilibrium and Composition of Forces Which Act on a Material Point.

Section 1: On motion and force, also on the composition and resolution of forces

ยง 1.] A body appears to us to be in motion, when it changes its relative situation with respect to a system of bodies supposed to be at rest; but as all bodies, even those which appear in the most perfect repose, may be in motion; a space is conceived of, without bounds, immoveable, and penetrable by the particles of matter; and we refer in our minds the position of bodies to the parts of this real, or ideal space, supposing the bodies to be in motion, when they correspond, in successive moments, to different parts of this space.

The nature of that singular modification, by means of which a body is transported from one place to another, is now, and always will be, unknown; it is denoted by the name of Force. We can only ascertain its effects, and the laws of its action. The effect of a force acting upon a material point, or particle, is to put it in motion, if no obstacle is opposed; the direction of the force is the right line which it tends to make the point describe. It is evident, that if two forces act in the same direction, the resultant is the sum of the two forces; but if they act in contrary directions, the point is affected by the difference of the forces. If their directions form an angle with each other, the force which results will have an intermediate direction between the two proposed forces. We shall now investigate the quantity and direction of this resulting force.

For this purpose, let us consider two forces, \(x\) and \(y\), acting at the same moment upon a material point \(M\), in directions forming a right angle with each other. Let \(z\) be their resultant, and \(\theta\) the angle which it makes with the direction of the force \(x\). The two forces \(x\) and \(y\) being given, the angle \(\theta\) and the quantity \(z\) must have determinate values, so that there will exist, between the three quantities \(x\), \(z\) and \(\theta\), a relation which is to be investigated.

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