MECHANICS. 



168 



Fig. 23. 



at X,. But then, thoro is tho resistance of tho table at o,, acting 

 upwards to prevent that motion ; and consequently the body 

 rrm:um :tt rr-t, or i.i in equilibrium. 



this hUiloinriit holds equally good when tho piano on 

 which the body rests ia slop, d t,i- inrlin.-d to tin- hori/.onUl 

 piano; us is rvid.-nt from Fig. 23, whore tho cylindrical body 

 on HID slope A it must upset if o i* folia outside tho base x Y z. 

 Wo may, tlHTrfoiv, fondudo generally both aa to hori/ontul 

 and inclincd planes that a body will rest in equilibrium on u 

 l>huio, if tho vortical lino, passing 

 through its crnlro of gravity, moots 

 tho piano within tho base. If it 

 meets it outsido tho base, the body 

 will overturn. 



Between these two, it should be 

 observed that there is an interme- 

 diate case, in which tho perpendicu- 

 lar meets tho piano neither within 

 nor without tho base, but on its 

 circumference. When thia happens, 

 tho body ia equally disposed to 

 stand or upset ; but, in fact, it will 

 overturn ; for in, such an unsteady 

 position the slightest touch or shako 

 would send it over. It is a case of unstable equilibrium. 



In interpreting and applying this principle to practice, you 

 must be on your guard as to the meaning of the word " base ; " 

 else you may imagine some day you have discovered that a body 

 does not upset when tho vertical from the centre of gravity 

 falls outside the base. Suppose the base to be bent inwarda 

 into a horse-shoe form, as in the cone, a (Fig. 24), or into the 

 form of the semi-circular wall, I, in which latter case tho centre 



of gravity is with- 

 out the substance 

 of the body ; then 

 the point p is on 

 the floor, outside 

 the spaces along 

 which the bodies 

 are in contact with 

 it. Still, neither 

 body will ups-ct ; for 

 the advanced spurs 

 Fig. 24. of the bases at Tf 



and z will act aa 



props, and in order to upset they must turn over the lino 

 T z joining them. This shows that the real base includes 

 all the open space within T z; and you learn that, when- 

 ever tho base of contact bends inwards, yon must measure 

 the base of support from one projecting point to another 

 all round, making no account whatever of the inward bends. 

 A common table touches the floor only at four points, and a 

 round table at three ; but in both the base of support is 

 all the space within the oblong or triangle got by joining these 

 points. 



There ia another class of cases to be noticed, those which are 

 round all over their surface like a ball, or egg, or sea-shore 

 pebble, and have no flat bases + rest on that is, which can be 

 supported at only one point of their surface ; or, where there are 

 hollows on them, along a line of points surrounding the hollow. 

 This latter case we 

 need not consider, for 

 such bodies rest, like 

 those we have already 

 examined, so far as tho 

 hollows are concerned =?-- 

 (as in d, Fig. 25), on 

 wide bases. 



Confining attention, 

 therefore, to cases in 

 which thera aro no Fig. 25. 



hollows, or the surface 



is convex all round, if you place such a body, say an oval, 

 in tho position represented it a (Fig. 25), the perpendicular, 

 o p, from its centre of gravity, G, on the plane will fall out- 

 Bide ita base, or point of support, o, and it will roll over 

 until, after rocking for a few turns, it settles into the position 

 o, in which a is above o. Now move it further from thia until 



it roaches tho position c, in which again a will be over the 

 point of support, o ; and again yon will have a pouible equi- 

 librium, that JH, ponhiblo in imagination, for tho body it supported 

 from below. But actually to produce equilibrium in thU caso U 

 tho celebrated problem of Columbus, which that great narigator 

 solved after BO summary a fa "mteady i* It, that the 



body drops immediately into the poxition '/. 



Of this unsteady, or unstable equilibrium, we shall hare more 

 in tho next lesson ; my object hero ia to point out the fact thtt 

 in both positions, I and c, the line a o ia perpendicular to the 

 surface of the body. It in evidently perpendicular to the plane 

 on which the oval rests ; but, since the hitter's surface touches, 

 or coincides at o, with that plane, u O must be perpendicular 

 also to that surface. Hence wo learn that, whatever be the 

 number of points at which a convex body can rest, steady or 

 unsteady, on a horizontal plane, for every one of these points 

 the lines connecting them with the centre of gravity must pierce 

 its surface at right angles ; or 



The number of positions of equilibrium of a convex body, 

 supported on a horizontal plane, is equal to that of the perpen- 

 diculars to its surface which can bo drawn from its centre of 

 gravity. 



A few instances in illustration of the principles explained in 

 this lesson will now be useful. When a man stands upright, the 

 base by which he is supported ia ao much ground under him as 

 is covered by his feet, together with the space between them. 

 If he widens that space to left and right, he makes himself more 

 steady as to being thrown sideways, but ia more easily cast on 

 his face. If he puts ono foot before the other, he become* 

 steadier at front and back, but less so to his sides. A two- 

 wheeled gig, or Hansom, to bo properly balanced, should have 

 its centre of gravity over tho lino joining the points at which 

 the wheels touch the ground. If it be in advance of that line, 

 it will throw a weight on tho horse's back ; if behind it, the gig 

 will upset backwards should the 

 belly-band break. 



A body may be made to roll up 

 an incline by loading it at one 

 side. Take a round ball of cork, 

 for instance, and put some lead 

 into a hollow scooped out near its 

 surface, closing the hole so as to 

 leave the ball perfectly round. 

 The centre of gravity will then 

 no longer be at the centre of the 

 ball, but to one side, let it be at 

 o (in a, Fig. 26). Put the ball now on the incline, with the 

 leaded side looking up the slope ; the perpendicular o P will 

 meet the incline above o, and the ball will continue to roll 

 upwards until the centre of gravity, o, comes over the point oi 

 support. 



This experiment may be tried in another form without the 

 use of the lead, by simply scooping a hollow on one side, or as 

 in the following example: Get a round cylinder of cork a 

 common bottle cork and scoop out its substance on one side, 

 as represented at 6, Fig. 26, preserving carefully the roundness 

 of the two circular faces at its ends. Put this cylinder on the 

 incline, with the scooped part facing down the slope, and you 

 will find that it will also run upwards, as did the ball. The 

 reason of thia yon will easily discover, but one word of caution 

 may be given : be careful, in making the experiment, that the 

 incline be not great. 



The following are other common instances of the principles 

 enunciated, which you can try aa 



EXAMPLES FOE PRACTICE. 



1. A man walking up a hill stoops forward; why ? And why, also, 

 when coming down, does he leau backward ? 



2. A person rising from a cbair leans his body forwards, and drawi 

 his feet close to the chair ; why ? 



3. Carrying a bucket of water with one hand, be leans to the side 

 opposite. 



4. Why does a corpulent person generally hold bis head up and 

 throw his shoulders backward ? 



5. A horso and rider are more apt to fall coming down a hill than 

 on the level road ; why ? 



6. An omnibus, or conch, is safer for trarelling irhen it is well filled 

 inside, than when outside. 



Fig. 26. 



