MECHANICAL PU1LOSOPHY.-STATICS. 



[THB MAOIC CLOCK. 



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If the body rest upon a single point, it is evident that 

 the pressure on this point is equal to the weight of the 

 bodj 



When the body rests upon two points A and B, as in 

 Fig. 78, where the body n* r*. 



is supposed to consist of 

 a circular disc and a 

 cylindrical stick, fixed 

 perpendicularly to its 

 centre, the pressures on 

 these point* may readily 

 l- .i. -;u, .n. i. 



Let O be the centre of 

 gravity of the body rest- 

 mgon the plane. 



rhi* bodV is evidently kept in equilibrium by its 

 weight, and the equal and opposite reactions produc< 

 by the two pressure* on the points A and B. 



The vertical passing through G must therefore pass 

 through the line adjoining A and B. 

 Let C be the point where the vertical cuts A B ; the 

 weight of the body acting a 

 G may be transferred from 

 GtoC. 



Then (Fig. 79) we have two 

 reactions, P, and P 2 , acting 

 upwards, perpendicular t< 

 the plane at A and B ; anc 

 a weight W equal to the 

 weight of the body acting downwards, and also perpen- 

 dicular to tho plane. 



Hence, Prop. XI., P .-J-Pj-W, and P 2 'B C=P 1 'AC. 

 Two equations are thus obtained from which P, and 

 P, may readily be determined, when the position of the 

 centre of gravity G of the body is known. 



The reactions P, and P a will be equal and opposite to 

 the pressures produced by the body at A and B. 



When the body rests upon three points A, B and C 

 (Pig. SO), we may still find the pressures produced by 

 its weight on these points. 



Join the points A, B and 0. In order that there may 

 tig. to. be equilibrium, the vertical passing 



through G, the centre of gravity of 

 the body, must fall within the 

 triangle ABC. (Prop. XXI.) 



Let C* be the point where the 

 vertical passing through C cuts the 

 triangle A BO. 

 C Join A C ; and produce A C" to 



meet B C in D. 

 If W be the weight of the heavy body, we may transfer 

 the force W from G to C. This force, acting perpen- 

 dicular downwards at C, may be resolved into two forces, 

 W, acting at A, and W, acting at D, both perpendicular 

 to the plane of the triangle ABC. 



Where W, + W, JW, and W, -A C-W..C D. 

 The force W, acting at D may again be resolved into 

 two other*, W. acting at B and W. at C, both perpen- 

 dicular to the triangle ABC. 



Where W. + W.-W,, and W,-BD=\WCD. 

 From these equations the forces W,, W,, and W,, 

 which will be the pressures exerted by the heavy body 

 on the points A, B and C, mav be readily determined. 

 If the heavy body rest on the plane by four points of 

 n.I. support, as a table on its 



four legs, A, B, C, D, 

 (Fig. 81), the pressures upon 

 these points will be inde- 

 terminate if we consider the 

 body perfectly ri 



since it may bo sup- 

 ported by any three of the 

 I -.ints of contact, the pres- 

 sure on the fourth may be either nothing or some hnite 

 proportion of the weight of the body. The method used 

 in tbo two preceding esses to determine the pressures 

 will fail in this. Toe only condition we have is, that 

 the cam of the pressures on the points A, B, ( 

 mast bs equal to the weight of the body ; and if we 



consiil.T them OH weights act.- lii'ularly t" tlm 



plane on In. -h the, table rests, tli" ivntiv uf J_T.-I\ ity f 

 these weights will lie in the 

 the centre of gravity of the table. 



The same reasoning may be extended to the cases 

 h. iv the points of contact are more than f<>ur. 



THK MAGIC CLOCK. The influence of the centre 

 of gravity on the position of equilibrium of a heavy 

 body, which can only turn round a fixed horizontal axis, 

 is well illustrated by an ingenious contrivance called the 



A transparent glass dial-face has a hole pierced thr. m^li 

 its centre at C (Fig. 82) ; in thin is a fixed horizontal 

 axis ; on this axis is placed the hour-hand of the clock, 

 which can move freely round it to the rijjht or left. 

 The extremity of the hand opposite the pointer 1! is 

 terminated by a hollow ring ; in this ring there is a heavy 

 spherical ball A, capable of moving freely round it. 

 This ball is made to move uniformly round the interior 

 of the ring, by a watch-movement concealed in the h.-uxl, 

 once in twelve hours, in the direction indicated by the 

 arrow in Fig. 82. The weight of the ball A is so pro- 

 rig. 8J. portioned to that of the hand 

 with its concealed watch-work, 

 that as A moves round thr interior 

 of the ring G, the common centre 

 of gravity of the ball and hand 

 would describe a circle round tho 

 centre of the axis C, if the hand 

 were fixed in one position. Sine,. 

 the hand can move freely round 

 C, for every new position of A 

 the hand will assume that position 

 which shall make C G vertical 

 (Prop. XIX.) As A, therefore, 

 is carried round the ring uniformly once in twelve hours, 

 the pointer of the hand B moves with the same unifor- 

 mity round the dial in the opposite direction, and in- 

 dicates the hour. 



The ball A, and the watch-movement which caus< 

 to turn round the ring, being both concealed, tho hand 

 seems to move by itself, as if by magic. The remarkable 

 property that the hour-hand being made to move by 

 your finger backwards or forwards, when left to ii 

 will, after a few oscillations, resume its position, and 

 wint to the correct hour, adds considerably to the 

 llusion. 



Let D (Fig. 83) represent the centre of the ring round 

 which the spherical body circulates ; 

 A,,, the centre of the spherical 

 body ; C, the centre of the axis 

 nmnd which the hand i. 

 B, the centre of gravity of the 

 hand and its concealed watch-work, 

 exclusive of the movable sphevi- 

 cal body; G 2 , the common ccntru 

 of gravity of the hand and spheri- 

 cal body. Also let A\ 

 the weight of the hand and works 

 exclusive of the sphere, whose 

 weight U represented by W. 



Then, if W and W be so chosen 

 that W'BC = W'-CD, the com- 

 mon centre of gravity <!... of tho 

 hand and movable sphere will de- 

 scribe a circle round C, if the 

 hand bo supposed to be ti\nl 



Fig. 85. 



while A 3 describes a circle 

 round D. 



Let A, represent the p< 

 of the centre of the sphere, when 

 it lies in U C D produced ; <! , the 

 centre of gravity of tho hand and 

 sphere for this position. 



Join D A,, C G 3 , and B G a A 2 . 



When tho centre of tho mov- 

 able sphere is in the p,,-in,,u A,, 

 we have a weight \\ 

 another W at Ai, in directions parallel to each other ; 



