248 



THE POPULAR EDUOATOE. 



MECHANICS. VII. 



AXIS OF SYMMETRY STABLE AND UNSTABLE EQUILIBRIUM 

 INTRODUCTION TO THE MECHANICAL POWERS, ETC. 



AXIS OF SYMMETRY. 



THERE is a large number of cases in which, though we may 

 not be able actually to find the centre of gravity, we can say it 

 is on some line in reference to which the body is symmetrically 

 formed. In an egg, for example, the line joining the round and 



pointed ends 

 is an axis of 

 symmetry. If 

 we make 

 cross sections 

 of it perpen- 

 dicular to this 

 line, they will 

 be all circles 

 through the 

 centres of 

 which the line 

 will pass. 

 The elliptic 

 oval at a, 

 Fig. 33, and 

 Fig. S3. the cylinder 



at c, and tho 



right cone at d, are instances. The cubical box at e, is another 

 in which tho cross section is a square, the line joining the meet- 

 ings of the diagonals on the upper and lower 

 faces being a symmetrical axis. The oval 

 board at b, also, in which all the dotted 

 lines are bisected by the arrow perpen- 

 dicular to them, is another instance, the 

 arrow being the axis of symmetry. Wher- 

 ever two such axes exist, of course the 

 centre of gravity is their point of inter- 

 section ; but if there be one only, as in 

 the portion of the ring in Fig. 34, the 



Fig. 34. 



tt 



position of the centre on it must be ascertained by other means. 



STABLE AND UNSTABLE EQUILIBRIUM. 



In the last lesson, I showed you that when a body rests in 

 equilibrium on a horizontal plane, the perpendicular from the 

 centre of gravity falls within its base. This condition being 

 satisfied, it will not upset of itself, but may be overturned from 

 without by a force acting sideways. What are the conditions 

 on which depend the ease, or the difficulty, with which it can be 

 Bo upset? Let three cylinders, a, b, c, Fig. 35, be taken in 

 illustration; the first of broad base and small height, the other 



two of equal heights and 

 bases, the latter narrow in 

 each. Suppose that a force, 

 say of one pound, repre- 

 sented by the dotted arrow 

 pointing to the right, is ap- 

 plied transversely to. each, 

 and let the weights of the 

 p bodies be represented by 

 6 arrows pointing downwards 

 on the vertical lines in 

 which their centres of 

 gravity lie. Now, the 're- 

 sultant in each cylinder of 

 these two forces, repre- 

 sented by the arrows slant- 

 ing to the right, is the up- 

 setting force. If this arrow 

 brikes_ the ground outside the base of any cylinder, it will over- 

 turn ; if within, it will remain standing as before. 



1. Now, taking any one of the cylinders, say a, it is evident 

 that the transverse force remaining the same, and the height at 

 which it is applied the same, the greater its weight is tho 

 longer will the arrow o p be, and therefore the more will the 

 resultant o R elope downwards towards o p, tending to fall 

 within the base. Therefore, everything else being the same, the 

 greater the weight of the body the less easily is it upset, that is, 

 we more stable it is. 



Fig. 35. 



2. Again, supposing the weights of the two cylinders a, c, to 

 be equal, but the base of the former greater than that of the 

 latter, if equal transverse forces, be applied to both at equal 

 heights, then o R being also equal in both and equally inclined 

 to o p, the resultant will tend more to fail within the base in a 

 than in b., that is, everything else being the same, the broader 

 the base, the greater the stability. 



3. Further, if, as in b and c, the bases and weights being the 

 same, and the transverse force applied to each cylinder being 

 still one pound, the force is applied higher up in one cylinder 

 than in the other, then the resultant is more likely to meet 

 the ground within the base in the latter than in the former ; 

 that is, the lower down the transverse force is applied, every- 

 thing else being the same, the greater the stability. 



4. Lastly, as is evident from d, e, /, in Fig. 35, when the 

 bodies incline to one side, the perpendicular from the centre of 

 gravity meets the base nearer to its circumference on that side ; 

 and, if the transverse force is applied in that direction, the 

 resultant tends more to fall outside the base ; that is, everything 

 else being the same, the stability is least when the upsetting 

 force acts in the direction in which the body leans. 



These are truths known to everybody from experience, but of 

 which here you see the "reason why," and what is of no less 

 advantage, you obtain a rule by which you may measure the 

 amount of stability or instability in any case that may come 

 before you. If you draw figures for bodies of different weights, 

 different bases, different transverse forces, and their heights of 

 application, you ,-\ 



will by trial feel /' 



your way, and / 



soon clearly un- "j 



derstand the 

 subject. 



But the casea 

 to which the 

 terms "sta- f 

 bility " and " in- .^ 

 stability " are f 

 more commonly 

 applied, are *** 

 those in which 

 there is only one 

 point of support, 

 and the slightest 

 force from with- 

 out causes dis- 

 turbance. In 



Fig. 36, as was shown in Lesson V., (page 188) the body 

 supported at the point o is in equilibrium in the two positions 

 o A B and o c D. Now the first of these is one of stability, 

 the second of instability. What do these terms denote P 

 This ; that, if you pull the stable body out of its rest into 

 any other position to right or left, say o E F, back it will 

 return to A o B, as though by a free choice. In the dis- 

 turbed position o E F, the weight acting downwards at o pulls 

 it back ; it can descend, but not ascend. Try the same on the 

 position o c D ; the body, no longer supported from below, can- 

 not re-ascend ; down it will rush to the stable position ; and, 

 after oscillating there for a few turns, come to rest. We see 

 thus that in stable equilibrium the centre of gravity is in the 

 lowest possible position ; in unstable in the highest. 



!?ow take the same body attached to the post at its centre of 

 gravity, G, Fig 37. However you turn it round, G is sup- 

 ported, and the body rests. The equilibrium, therefore, is 

 neither stable nor unstable. It 

 neither returns on disturbance 

 to the first position nor rushes 

 away from it. This is termed 

 " neutral equilibrium ; " the f|j 

 centre can neitlier ascend nor 

 descend. 



Now take the e 

 bodies, Fig. 38 : that represented ; 

 at b is stable, for the centre of 

 gravity, supported from below, is 

 in the lowest possible position. Disturb it into the position, a, 

 this centre ascends, and the weight pulling downwards brings it 

 back to b. The body in the position c is unstable. It is in 



Fig. 37. 



Fig. 38. 



