where v=0, 88, where 8 is length of curve OB.

8=

Solving for the velocity, we have

it represents the relation between distance and time. When 8 = 8p, tt, so that

therefore, c=tr•

It is evident that s is a periodic function of the time, and that it repeats itself at intervals of time t, such that

t = 2 = √⋅

This value th represents the time taken by the body from leaving the position B until its return. One half of this

B

vào

In general, when the angle is not small, the equation - g sin ads becomes, since ds sin α = dy,

Svdv=-9 S'dy.

Integrating, we have v2=-2gy2g(h-y).

It will be seen that this result is the same as if the body had fallen freely through a height hy (see Art. 74). The value for v is evidently true whatever be the vertical curve in which the body moves, providing the only forces acting on the body are the force of gravity and another force normal to the curve. The foregoing fact leads to the statement, in descending along any curve without friction, from a height h to any other height y a body will have the same velocity as if it fell freely through the height h This fact is often made use of in mechanical problems. The summation of forces normal to the path gives

y.

T-G cos α=

Mv2
ī

Mv2

and, therefore,

TG cos a +

which gives a value for the tension in the cord.

Since v

is greatest when a = 0, it is seen that the greatest tension in the cord occurs when the pendulum is vertical.

If now we make use of the fact that the body moves in a circle whose equation is x2+ y2 - 2ry = 0, and remember that ds2dx2 + dy2 for any curve, we may write

√2ry - y2. √2g(h− y)

The integral of the expression on the right-hand side of the equation is not expressible in terms of ordinary algebraic or trigonometric functions, but must be expressed in terms of the elliptic functions. The student may not be familiar with such functions, so that we shall express it approximately by means of an infinite series. This series will be sufficiently rapidly convergent if the radius of the circle is large and the distance OB is small. Using the minus sign in the numerator, since t is a decreasing func

tion of 8, we may write

For very small values of h we may neglect the terms containing h. The result then becomes

t = T1

since rl, and this is the same result that was obtained before.

This means that for small values of a, not greater than 4o, the time of vibration of a simple circular pendulum is a constant; that is, the oscillations are isochronal.

It is seen that the time of vibration of a simple pendulum varies as the square root of its length, for any locality on the earth. In order to get a pendulum that will beat seconds it is necessary to place t=1. Knowing the value of g for the locality, the proper length may be determined. If we measure the length of a pendulum and its period we may calculate the value for g for any locality. This is the easiest and most accurate way of determining g.

A

FIG. 112

Problem 128. The centrifugal railway (Fig. 112), or "loop the loop," is a common example of a simple circular pendulum, where the effect of the string is replaced by a track. If we neglect friction, the only forces acting on the car are the force of gravity and the

normal pressure of the track. Suppose the car starts from rest at a height, h. What must be the relation between h and h', so that the car will pass the point A without leaving the track.

HINT. The velocity at the lowest point, v2 = 2 gh, is the same as the velocity with which the car comes down. The centrifugal force must be great enough at A to overcome G, the weight of the car (h=‡h').

Problem 129. In the simple pendulum find the value of y in terms of h for which the tension in the string is the same as when the pendulum hangs at rest.

Problem 130. A pendulum vibrates seconds at a certain place and at another place it makes 60 more vibrations in 12 hours. Compare the values of g for the two places.

89. Cycloidal Pendulum.-It has been found that a pendulum may be obtained whose period of vibration is

constant by allowing the string to wrap itself around a cycloid as shown in Fig. 113. The pendulum hangs from the point A. AB and AC are cycloidal guides around