Proposition 99

PROPOSITION 99. Problem

896. Pendulums are set in motion in rotational motion ; to determine the motion and the curved line described on the surface of a sphere.

Solution

Since the moving body is bound by the pendulum, it moves on the surface of a sphere of which the radius is the length of the pendulum. Let this length or the radius of the sphere be equal to a ; from the nature of the circle AM (Fig. 96 and Fig. 97) it follows that z = a − (a 2 − u 2 ) . Hence we have and thus

From these is found the centripetal force attracting towards A, which put in place, in order that the body can move freely in the projection BQC , is equal to And this equation is obtained for the curve BQC : which is sufficient for the construction of the projection BQC. Let the tangent at B be perpendicular to the radius AB, which must always happen somewhere, unless the projection is a straight line, since the centripetal force decreases with decreasing u. Putting AB = f; then i = a − (a 2 − f 2 ) and [p. 497] If besides it should be that

then the body is rotating in a circle, the radius of which is f, with a speed corresponding to the height

The length of the pendulum completing whole oscillations in the same time is equal to (a 2 − f 2 ) . But if it is not the case that 2c3 = gf 4 (a 2 − f 2 ) , and now the difference is very small, then the curve BQC does not depart much from the circle. In order that the positions of cusps of this curve can be found, just as in Proposition 91 of the preceding book : Hence this becomes : and Since the curve is nearly a circle, put u = f and with which done the is produced :

Hence it follows that the interval between the apses is the angle :

Clearly the position for an angle of such a size, at which the pendulum is at a maximum distance from the axis, is placed at intervals with the position at which the pendulum is closest to the axis. Q.E.I.

Corollary 1.

897. Therefore in order that the pendulum AB = a (Fig. 98) describes the circle BCDE in g .BO 2 a rotational motion, it is necessary that the speed corresponds to the height 2.AO .

Corollary 2.

898. Now the length of the pendulum, that completes the smallest whole oscillations in the same time, in which the period is completed in the circle BDCE, is equal to AO.

Corollary 3. [p. 498]

899. Therefore the times, in which different circles are traversed by the rotating pendulum AB, are in the square root ratio of the heights AO.

Corollary 4.

900. Therefore since the pendulum of length a makes a maximum horizontal circle, the radius of which is a, an infinitely great speed is required and each period is completed in an infinitely short time.

Corollary 5.

901. If the radius of the circle BO should be very small with respect to the pendulum AB = a, the periods of the rotational motion are in agreement with the whole oscillations of the same pendulum.

Corollary 6.

902. If the curve described by the pendulum should not be a circle, but a close figure and BO is very small, then the angle between two apses is 900 or a right angle.

Corollary 7.

903. Now in this case the curve described by the body is an ellipse having the centre at A. Which can be gathered from the centripetal force, which then becomes proportional to the distances.

Corollary 8

904. Moreover since the larger the radius BO becomes, from that also the greater becomes the angle intercepted between the two apses. And on making BO = BA, the angle here is 1800.

Corollary 9.

905. If the angle BAO is 30 degrees, then BO = 12 BA or f = 12 a . Therefore the angle intercepted between the two apses is 360 degrees or 990 50’. Hence this figure 13 abcdefghik, etc is of the projection in the horizontal plane (Fig. 99), in which the highest apses are at a, c, e, g, i and the deepest at b, d, f, h, k.

Corollary 10.

906. Therefore in this figure the line of the apses is moving in succession; for each period progresses by almost 390 in succession.

Corollary 11.

907. But if that angle BAO should be less than 30 degrees, then the progression of the apses is less also. For whatever the angle BAO which is known at some point, I resolve the fraction in a series, which is the following : Hence in one period the line of the apses is moved forwards by the angle 135 f 2 degrees a2 as an approximation, if f is very small.

Corollary 12.

908. From these it is apparent that the movement of the line of the apses in individual periods is almost in the square ratio of the sines of the angle BAO.