Page:Elementary Text-book of Physics (Anthony, 1897).djvu/68

54 velocity about the points $$a, a_{1}$$, and the axle of the wheel will remain horizontal.

Let Fig. 20 represent the rotating wheel of the former diagram, the axis being supposed to be nearly horizontal. If a weight be hung at the point $$e$$, it tends to turn the wheel about a horizontal axis $$CD$$. The direction of motion of the particles at $$A$$ and $$B$$ is not changed by this rotation, but the particles at $$C$$ and $$D$$, and to a less extent all the other particles on the rim of the wheels are forced to change their directions of motion. Now it has been shown (§ 27) that the change in the direction of motion of a particle is equivalent to a force $$\frac{mv^2}{r}$$ where $$m$$ is the mass of the particle, $$v$$ its velocity, and $$r$$ the radius of the circle in which, it moves. The reaction of the particle is directed outward along the normal to the curve; in the case of the particles considered at $$C$$ and $$D$$, this reaction is directed to the right at $$C$$ and to the left at $$D$$. These two forces, therefore, and all others like them due to the reactions of the other particles, combine to form a couple which tends to rotate the wheel about the axis $$AB$$. This rotation about $$AB$$ gives rise to similar reactions at $$A$$ and $$B$$, the reaction at $$A$$ being directed to the left and at $$B$$ to the right. These forces, and all other similar ones arising from the other particles of the wheel, combine to form a couple which tends to rotate the wheel about the axis $$CD$$ in the opposite sense to that in which it is rotated by the weight at $$e$$. Thus the weight applied at $$e$$ will produce a rotation about the vertical axis $$AB$$.

48. Central Forces.— We now turn to the consideration of the motion of a particle acted upon by a force always directed toward a fixed centre or a central force. Its motion will exhibit one peculiarity which is independent of the law of the central force. The radius drawn from the centre to the particle will always sweep out equal areas in equal times, whatever be the law of the force.

It is at once obvious that the motion of a particle, acted on by