which are the accelerative force and pressure due to the angular velo

city respectively,

Again, the accelerating force down the curve due to gravity is

Ex. Let the curve be the quadrant of a circle convex to the axis of rotation, which also touches it,

which gives the velocity at any point of the descent, and solves the problem.

To find when the ring would fly off from the circle if unrestrained, or when the pressure changes its direction, we have

G.) ·472

The student will find no difficulty in making other applications.

619. Let y denote the ordinate of the generating parabola, on the radius of the circle in which the body revolves; then since the force of gravity is counteracted by the centrifugal force and the reaction of the surface, we get by the triangle of forces the centrifugal force, which in a circle is equal to the centripetal one, expressed by

CONSTRAINED MOTION.

p being the parameter of the parabola. Hence the time of revolution is (see 440)

Pis

2

But the time of oscillation of a pendulum whose length is

The pressure against the surface is easily found to be

620.

Let a

be the distance fallen through to acquire

the velocity, a being the altitude of the cone; then if generally the equation of the generating ▲ be

α

yx tan. a

being the inclination of the slant side to the axis, we have

v = √2g. (a + x) = √ Fy,

F being the centrifugal force. But since gravity is counteracted by Fand the pressure against the surface,

F= g. 2.

y

a remarkable result, inasmuch as it shews that the altitude due to the velocity with which the body is whirled is the same for all cones of the same altitude. Hence

621.

Let R be the radius of the earth, x any latitude, t the time of the earth's rotation, and W, W' the weight of the same body at the equator, and in latitude a respectively; then at the equator the centrifugal force is

and the resolved part of o' which counteracts gravity is therefore

622.

If L denote the length of the pendulum, F the force, and F the time of an oscillation, then (see Venturoli, or Bridge,)

623.

Ifr altitude of the point descended from, and h'

that of the point descended to, r being also the radius of the generating circle; then the time of descent is

whence 4r, the length of the pendulum, may be found by approximation.

624.

Let P and p be the lengths of a degree at the pole and equator, m, n, the lengths in latitudes, x, x'; then since the length of a degree a radius of curvature, we have