64 LECTURE IX. 



which have been already mentioned, some have considered the square of the 

 velocity as affording the true measure of force ; but the properties of 

 motion, concerned in the determination of rotatory power, are in reality 

 no more than necessary consequences of the simpler laws on which the 

 whole theory of mechanics is founded. 



The effects of rotatory motion may be very conveniently examined, by 

 means of an apparatus similar to that which was employed for the same 

 purpose by Mr. Smeaton.* A vertical axis is turned by a thread passing 

 over a pulley, and supporting a scale with weights ; the thread may be 

 applied at different parts of the axis, having different diameters, and the 

 axis supports two arms, on which two leaden weights are fixed, at distances 

 which may be varied at pleasure. The same force will then produce, in 

 the same time, but half the velocity, in the same situation of the weights, 

 when the thread is applied to a part of the axis of half the diameter : and 

 if the weights are removed to a double distance from the axis, a quadruple 

 force will be required, in order to produce an equal angular velocity in a 

 given time. (Plate V. Fig. 74.) 



When a number of connected bodies, or a single body of considerable 

 magnitude, is made to revolve round a centre, it is sometimes necessary to 

 inquire into what point their masses might be supposed to be concentrated 

 so as to preserve the same rotatory power with the same angular velocity. 

 This point is called the centre of gyration. In a circle, or any portion of a 

 circle, turning round its centre, the square of the distance of this point from 

 the centre, is half the square of the semidiameter ; and the whole effect of 

 the momentum of the circle upon an obstacle at its circumference, is exactly 

 half as great as that of an equal quantity of matter, striking the obstacle 

 with the velocity of the circumference. 



There is another point, of which the determination is of considerable 

 utility in many mechanical problems : this is the centre of percussion ; or 

 the point at which an obstacle must be applied, in order to receive the whole 

 effect of a stroke of a body which is revolving round a given centre, with- 

 out producing any pressure or strain on the centre or axis of motion. In 

 a straight line, or a slender rod fixed at one extremity, the distance of this 

 point from the centre of motion is two thirds of the whole length.t 



The same point is also the centre of oscillation, the distance of which 

 determines the time of oscillation or vibration of the body, suspended as 

 a pendulum upon the given centre of motion.^ It may easily be shown 

 that a rod a yard long, and of equable thickness, suspended at one ex- 

 tremity, vibrates in the same time as a ball suspended by a thread of which 

 the length is two feet. But if the rod were suspended on a centre at some 

 point within its extremities, the time of its vibration would be prolonged, 

 so as to become equal to that of a simple pendulum of much greater length. 



* Ph. Tr. 1776, Ixvi. 450, and plate. See an examination of this paper in 

 Atwood, p. 382. 



t Lahire, Hist, et Mem. Paris, ix. 175. Parent, ibid. 1700, H. 149. Bernoulli, 

 ibid. 1703, pp. 78, 272, H. 114 ; 1704, p. 136, H. 89. Clairaut, ibid. 1735, p. 281, 

 H. 92. 



J Huygens, Hist, et Mem. de 1'Acad. x. 446, 462, and Hor. Osc. 121. John Ber- 

 noulli de Natura Centri Oscil. 1714. Taylor, Ph. Tr. 1713. 



