753 



MECHANICAL PHILOSOPHY II YDROSTATIC& [FLOMIXO BODTB. 



water : when placed in this mixture, the exciseman's 

 hydrometer sinks to the point marked proof upon 

 the scale ; if the liquor be below proof, or have 

 more than half water, the instrument will not sink 

 BO far as proof ; and if it be above proof, it will sink 

 further, and the surface of the liquor will stand above 

 proof. 



Uut in all determinations of specific gravity, account 

 must be taken of the temperature of the substance at 

 the time of the experiment : all bodies expand by heat, 

 and therefore become specifically lighter. Spirits more 

 especially are susceptible of this increase of volume by 

 increase of temperature : a cubic inch of brandy, for in- 

 stance, will weigh about ten grains more in winter than 

 in summer, unless the change of temperature is obviated 

 by artificial means. It is necessary, therefore, in 

 measuring the specific gravity of a liquid, to notice to 

 what extent the temperature of it differs from the 

 standard temperature of 60 Fahrenheit ; and then, from 

 knowing, by previous experiment, the amount of expan- 

 sion or contraction of the liquid for different degrees of 

 temperature, to reduce the specific gravity to what it 

 would be at the temperature of 60. 



We shall now give a few examples in practical 

 illustration of the preceding articles on specific gra- 

 vity. The following examples will be useful in connec- 

 tion with Applied Mechanics. 



Kxamplet. 1. If a piece of stone weigh 10 Ib. in air, 

 and only 6J Ib. in water, required the specific gravity of 

 the stone. Ans. 3-077. 



J A piece of elm weighs 15 Ib. in air, and a 

 piece of copper weighing 18 Ib. in air, and 16 Ib. 

 in water, is affixed to it ; the compound weighs 6 Ib 

 in water : required the specific gravity of the elm. 

 Ans. 6. 



3. A piece of cast-iron, of specific gravity 7 '425, 

 weighed 40 ounces in air, and 35-01 ounces in a 

 fluid : required the specific gravity of the fluid. 

 Ans. 1. 



4. Required the number of cubic inches in a block of 

 stone of specific gravity 2 '52, which weighs 1 cwt. Ans. 

 1228$ nearly. 



5. A composition of 1 cwt. consists of tin of specific 

 gravity 7 "32, and of copper of specific gravity 9 ; the 

 specific gravity of the composition is 8'784 : required 

 the quantities of tin and copper in the mixture. Ans. 

 100 ll>. of copper and 12 Ib. of tin. 



t>. The dimensions of one of the marble stones in the 

 walls of Uaibeck are, length 63 feet, and breadth and 

 thickness each 12 feet ; the specific gravity of the stone is 

 -7 : required the weight of the block. Ans. 683 tons 

 nearly. 



7. A solid, weighing 250 grains in air, weighs 147 

 grains in one fluid, and 120 in another : what is the 

 ratio of the specific gravity of the fluids 1 Ans. As 103 

 to 130. 



8. A cubical iceberg is 100 feet above the level of the 

 sea, the sides of it being vertical : given the specific 

 gravity of sea- water 1-0203, and of the ice -9214: re- 

 quired the number of cubic feet in the mass. Ans. 



ibie feet. 



'J. The weight of a common hydrometer is equal to 

 that of a volume v of water ; the part to which the 

 stem is attached weiglis a volume v' of water ; the 

 radius of the stem is r ; and the specific gravity of a 

 liquid, in which the instrument is immersed, is s : 

 required the length I of stem immersed when in equi- 

 libria. 



EQUILIBRIUM OF A FLOATING BODY. In 

 order that a body may float on a liquid and remain at 

 rest, two conditions are necessary : 



1. The weight of the body must be less than the 

 weight <if a volume of the lit. id <>f the HUIIIO bulk as 

 the body ; it must be equal to the weight of the volume 

 ilupUcod by that portion of the bulk which is immersed 

 iu it. 



2. The centre of gravity of the body, and the centre of 

 gravity of the fluid displaced by the partial iiniin 

 must both be situated in the same vertical straight lino 

 (P. 75:i). 



These are the only conditions necessary to secure the 

 Pig. 171 equilibrium of the floating body. 



If either of them be wanting, the 

 body (though it may not sink) will 

 move, and can come to a state of 

 rest only when these two conditions 



Suppose, for instance, a body 

 floating at rest on the water, as in 

 Fig. 172, the centres of gravity 

 of the body and of the displaced 

 fluid being in the same vertical line, 

 to be disturbed from its position, as 

 in Fig. 173. The fluid displaced 

 then becomes changed in form and 

 situation, though not in volume ; its centre of gravity 

 has shifted, and is no longer in the same vertical with 

 the centre of gravity of the body. The body, therefore, 

 is now acted upon by two forces Fig. 17*. 



tending to turn it round into the 

 position from which it has been 

 disturbed ; namely, the force due to 

 the weight of the body acting down- 

 wards through G, and the force due 

 to the pressure of the displaced fluid 

 acting upwards through G . 



When a body thus slightly disturbed 

 tends, as in this case, to return to its 

 original position of equilibrium, that 

 position is said to be one of ttable 

 equilibrium, in reference to the dis- 

 turbance in question. If the disturb- 

 ance were to act vertically on the body through its 

 centre of gravity, and therefore tending either to depress 

 the body lower in the fluid or to raise it further out ; in 

 the former case the additional upward pressure, and in 

 the latter the additional downward weight, would bring 

 the body into its original state of rest, the equilibrium 

 being always stable in reference to disturbance in a ver- 

 tical direction. 



Unstable equilibrium is that position of the floating 

 body, from wlu'ch, if disturbed, it yields to the disturb- 

 ance ; and instead of recovering its original condition of 

 equilibrium, departs from it more and more, till it finds 

 either some new position of equilibrium, or upsets and 

 sinks. 



When a floating body is slightly disturbed from rest 

 by being turned about its centre of gravity, the line 

 through that centre, originally vertical, becomes slightly 

 oblique ; the downward pressure or weight is then in the 

 direction of a new vertical line through the same centre ; 

 the upward pressure, too, in consequence of the change 

 of figure and position of the displaced fluid causing a 

 change in the centre of gravity of that fluid, is also in 

 a new vertical lino. The point, where this new vertical 

 meets the slightly oblique line through the centre of 

 gravity of the body, is called the mctacentre ; and the 

 vertical itself is called the line of support. We have 

 already seen that it is necessary for the line of support 

 to pass through the centre of gravity of the body when 

 the body is at rest (p. 753). 



The position of the metacentre, in reference to the 

 centre of gravity of the body when slightly disturbed, 

 determines whether or not the disturbance will be counter- 

 acted, and the body right itself by returning to its 

 former position of equilibrium ; that is to say, whether 

 or not this position of equilibrium is a ttable position. 

 If the metaceutre be at a point on the oblique line, 

 slightly turned out of its vertical position by the disturb- 

 . which is iibove the centre of gravity on that line 1 , 

 then it is plain the upward pressure, acting along the 

 il (line of support) through the metaceutre, must 

 re-establish the oblique line, or axis referred to, in its 

 . 1 .'or the centre of gravity is, as it 

 . a fixed point, round which the body turns ; and 



