Page:Elementary Text-book of Physics (Anthony, 1897).djvu/42

28 further statement that more than one force can act on a body at the same time, leads directly to a most important deduction respecting the combination of forces; for the parallelogram law for the resolution and composition of velocities being proved, and forces being proportional to and in the same direction as the velocities which they cause in any given body, it follows, if any number of forces acting simultaneously on a body be represented in direction and amount by lines, that their resultant can be found by the same parallelogram construction as that which serves to find the resultant velocity. This construction is called the parallelogram of forces.

In case the resultant of the forces acting on a body be zero, the body is said to be in equilibrium.

(3) When two bodies interact so as to produce, or tend to produce, motion, their mutual action is called a stress. If one body he conceived as acting, and the other as being acted on, the stress, regarded as tending to produce motion in the body acted on, is a force. The third law states that all interaction of bodies is of the mature of stress, and that the two forces constituting the stress are equal and oppositely directed.

27. Constrained Motion.— One of the most interesting applications of the third law is to the case of constrained motion. If the motion of a particle be restricted by the requirement that the particle shall move in a particular path, it is said to be constrained. If the velocity of the particle at a point in the path, at which the radius of curvature is $$r$$, be $$v$$, its acceleration toward the centre of curvature is $$\frac{v^2}{r}$$, and the force which must act on it in that direction is $$\frac{mv^2}{r}$$. However this force is applied, whether by a pull toward the centre or by a push or pressure from the body determining the path, or by the action of the forces which bind the particle to others moving near it, the reaction of the particle will in every case be equal to $$\frac{mv^2}{r}$$, and will be directed