174. A particle acted on by gravity moves in an arc of a vertical circle, to determine the motion.

Taking the vertical diameter as axis of y, and its lower extremity as origin, the equation to the circle is

if we suppose the motion to commence at the point defined by y,; and therefore

dt

dy

α

1

√(2g) √{(y,—y) (2ay — y2)}

(1).

If we put y=y, sin2 0, we have for the time of falling through any arc

an elliptic integral of the first order, whose value for given limits can only be approximated to; except when y1 = 2a, that is, when the velocity is that due to a fall from the highest point of the circle. This case we will soon consider (§ 176).

Suppose it be required to determine the time of descent to the lowest point; the limits of y are y, and 0. If we notice that

Hence the time of fall to the lowest point is

When the arc of vibration is very small, we have

The value of t, coincides with that in a cycloid, § 173, if we observe that in the cycloid the quantity a is 4 times as great as in the circle.

175. The next approximation gives, as a correction to the period of a quarter oscillation, the expression

whose ratio to that period is

=

Sa

(chord semi-angle of oscillation)2.

Thus, if the particle oscillate through an arc whose chord is 1 on each side of the vertical, the time of oscillation given

10

by the formula π will be incorrect by about

amount, in defect.

Ng

When the particle is supposed to be suspended by a thread without weight, it becomes what is termed a simple pendulum. Such a machine can exist only in theory, but Dynamics furnishes us with the means of reducing the calculation of the motion of such a pendulum as we can construct, to that of the simple pendulum. It is evident that by its means we may determine the value of g, if the length of the pendulum, its arc of oscillation, and the number of vibrations it makes in a given time, be known. Since gravity decreases (according to a known law) as we ascend above the Earth's surface, the comparison of the times of vibration of the same. pendulum on the top of a mountain and at its base would give approximately the height. One of the most important applications of the pendulum is that made by Newton. It is evident that if the weight of a body be not proportional to its mass, the value of g will be different for different materials. Hence the fact that pendulums of the same length vibrate in equal times at the same place whatever be the matter of which the bob is made, proves, by means of the above formula, the truth of one part of the Law of Gravitation, § 149: viz. that, ceteris paribus, the attraction exerted by one body on another is proportional to the quantity of matter it contains, and independent of its quality.

176. Or we may take the equation for the acceleration along the arc.

Suppose O to be the center, OA the vertical radius, B the point whence the particle starts with velocity aw, at time t=0; Pits position at time t..

Let AOB-a, AOP=0, OA = a.

i.e. unless the velocity of projection at B, be that due to a fall through the difference of altitudes of B and the highest point of the circle.

which determines the motion completely..

From the remark in § 171, it is evident that, after reaching A, the particle will ascend the other semicircle with a velocity just sufficient to carry it to the highest point; the time, T, at which it will reach that point after leaving A, will be found by putting

or, the particle will continually approach the highest point, but never reach it.

177. To find the pressure on the circle.

Suppose R directed outwards from the center, then evidently