

M0 



: am, hut en disturbance roll* through the po 



|M.-itii.ii '. In this case also you MO the centre, for 



in its /OKV.S/ iiontianf for instability, i" ^ 

 .:i<l balls, vuoh aa in l 



Fip. 40. 



Fig. 39. 



(page 220), oro nevii-'tl, 1'n-ir centres, as yon roll thorn on the 

 ground, can wither ascend nor descend. 



Tak.- :lla in n, Fig. 39, which represents a geological 



of hills and valleys. Those on the tops of the hills are 

 unstable, becauso thrir centres of gravity arc in their highest 

 positions. Disturb them, and down 

 they roll into stable positions in 

 the valleys, tho lowest positions of 

 those ci'iilrrs. But hero now a 

 new principle is brought to light. 

 A body may admit of several posi- 

 tions of equilibrium, but an unstable 

 'ays betu-een two gtables, and 

 a, stable between two unstables. 

 The ball in tho valley has a ball 

 perched on the hill on either side, 

 und the ball on tho hill has a ball 

 in the valley on either side. 



Take another illustration. Let 

 it bo a convex body, like a sea- 

 shore pebble, with one side, as in Fig 39, b, flatter than the 

 other. I showed you in the last lesson that such a body should 

 have as many positions of equilibrium on a plane as you can 

 draw lines from its centre of gravity piercing its surface at 

 right angles. Lot such points in this pebble be A, B, c, D, the 

 first and third more distant from the centre 

 o than the other two. If I now try to 

 make it rest on tho ground at A, the centre 

 being higher than it would be if the body 

 touched the ground on either side of that 

 point, it will roll down to either B or D, 

 which are two stable positions. We thus 

 learn that, 



The Positions of Equilibrium of a convex 

 body, supported from below, are alternately 

 stable and unstable. 



As a further illustration of the peculiarities of the centre of 

 gravity, take an egg. Why does it generally rest with its pointed 

 end downwards, as at d, Fig. 39, while an egg, as at c, turned 

 in wood of the same size and form, rests broad-end down ? Ex- 

 plain, also, the reason the prancing-horse toy, represented at 

 Fig. 40, supported at the edge of a table, 

 and having a wire attached to him, which 

 carries a heavy ball at its other end, does 

 not fall on the ground, but when disturbed, 

 rocks backwards and forwards. Also, how 

 a rocking-horse is set rocking by the child 

 on his back. The four-oared boat and 

 crew in Fig. 41, supported by tho point 

 of a needle on the iron upright below, imi- 

 tates a boat's motion at sea, rising, and 

 j-'unging, and going round, if the oars are 

 ioaded at their ends ; explain this. Also, 

 how the harlequin, Fig. 42, is balanced on 

 his pedestal, as he twirls round and bows, 

 leaning forward and falling backward at 

 the imminent peril of coming to the ground. Instances of this 

 kind could be multiplied without end, but as much as our space 

 allows has been said on the centre of gravity, which we shall 

 now leave to apply the principles BO far set forth to practice, 

 commencing with the Mechanical Powers. 



Fig. 41. 



Fig. 42. 



IXTBODOCnON TO THE MECHANICAL POWKKS. 



Before turning to the pif*"**"* 1 power*, the following prin- 

 iplo*. which are necessary to complete a knowledge of parallel 

 force* tho unit of them required for explaining the lerer nust 

 be established and understood. In the account given of parallel 

 force* in Le**on IV. mob only were considered M wet in the 

 tame direction, poll or push together, each adding to the effect 

 of every other ; and of thaw the subject of the centre of gravity 

 In Lesson* V. and VI. furnished numerous exemplification*, tie 

 force* all palling 

 towardstheearth's 

 centre. Now you 

 have to consider 

 twoforcefl.unequal 

 and parallel, bat 

 acting in ojtporite 

 direction*. 



Suppose two snob 

 applied to a body, 

 aa in Fig. 48, 

 where A and B are 

 the point* of ap- 

 plication, and the 

 arrows A F, B Q, 



represent their magnitudes and direction*. Let A P be 7 pounds 

 and B Q 3 pounds ; how can we find their resultant? From 

 a very simple consideration. Whatever it be, or at whatever 

 point it acts, it must be such that a force at that point, equal 

 and opposite to it, will balance it, and therefore make equili- 

 brium with its components A F, B Q. Now, that point cannot be 

 inside the line A B, for in that case the resultant of the two 

 which pull together could not be opposite to the third. The 

 point must, therefore, be outside A B and on the ride of the 

 greater force A P. Let the point therefore be o, and o B the 

 resultant, o B being the force equal and opposite to it, which 

 makes equilibrium with A F and B Q. 



Then, since there is equilibrium, the resultant of the two that 

 pull together, B Q and o s, most be equal and opposite to A P ; 

 and therefore, as proved in Lesson IV., A P is the sum of B Q 

 and o s. But A P being 7 pounds, and B Q 3 pounds, evidently 

 o s must be 4 pounds, the difference of these force*. The re- 

 sultant in, magnitude therefore is the difference of the com- 

 ponents. 



Now for the point of application. Since the resultant of 4 

 pounds at o and 3 pounds at B must cat B o at A inversely a* the 

 forces, if I divide A B into four equal part*, three of them will 

 be in A o ; or, which is the same thing, seven parts in B o and 

 three parts in A o, showing that o i* the point whose distances 

 from A and B are inversely as the forces. Putting all together, 

 wo learn that 



1. The Resultant of two Unequal Parallel Force* which act 

 at two points of a body in opposite direction* is equal in magni- 

 tude to their difference. 



2. Its point of application is outside of the greater force, 

 at distances from the points of application of the components, 

 which are inversely as these forces. 



The rule to be observed practically in finding this centre is, 

 to cut 'A B into as many equal parts as there are pounds, or 

 other units, or fractions of a unit, in the difference of the force*, 

 and then to measure outwards from A along the production of 

 A B as many of these parts as there are pounds or other unite in 

 B <J ; the point o so obtained is the parallel centre required. 

 And you see that what is thus proved for the number* 3 and 7 

 must hold equally for other numbers, whatever they be. 



There is one particular case of this principle, which I shall 

 just notice. Suppose A P become* equal to B q ; what of their 

 resultant ? how large is it, and where applied ? In magnitude 

 it is nothing, being the difference of the forces ; and the point of 

 application i* nowhere, at least within reach ; for on A B pro- 

 duced no point o can be found such that A o be equal to B o. 

 Pairs of forces of this kind are termed " couple*," and they play 

 an important part in Mechanics, in producing a tendency to 

 rotation ; but we shall not consider them here. 



One consequence more : How find the resultant of any num- 

 ber of parallel forces, some acting in one direction, others in the 

 opposite? Evidently by compounding separately, and finding 

 the centre* of, those which act in the opposite direction*. You 

 thus get two single parallel and opposite forces the reenltaota 



