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of two inclined planes, the height of each being half the thickness of the back of the wedge. When a wedge is driven into a piece of wood up to the head, the wood at each side is forced back as far as half the thickness of the head. This has been done gradually by the sides of the split being forced up the inclined planes formed by the sides of the wedge. Now, as a certain force balances a greater resistance on an inclined plane the longer the plane is, it follows that the longer the wedge used the greater is the power gained by using it, whether in splitting an object, or in raising a weight. Ships are raised in docks by driving wedges under the keel. Cutting and piercing instruments, as the plough, all act on the principle of the wedge. A familiar illustration of the principle is seen in the case of one glass tumbler placed within another, very little pressure on the uppermost one being sufficient to burst the lower.
We have seen (COHESION, p. 3) that all matter exists in one or other of the three states, solid, liquid, and gaseous. Liquids and gases have a general resemblance to each other in this respect, that their particles seem at liberty to glide about among one another without friction: they flow, and have hence received the common name of fluids, from Latin fluo, to flow. All liquids and gases have a certain degree of fluidity; but the property which chiefly distinguishes them is elasticity. A quantity of gas may be compressed into much less than its ordinary bulk, and when the pressure is removed it will return to its original volume; but no ordinary pressure produces any sensible compression on water or any other liquid. Liquids are thus practically incompressible, and therefore practically inelastic. The phenomena of liquids are of two kinds, corresponding to those of solids-the phenomena of liquids at rest, or in equilibrium, and the phenomena of liquids in motion. The department of Physics which treats of liquids at rest is called HYDROSTATICS, from Greek hydor, water, and statike, at rest. A few of the facts or laws in connection with this subject are now to be considered.
1. The fundamental principle of hydrostatics is, that when pressure is exerted on any part of the surface of a liquid, that pressure is transmitted to all parts of the liquid, and is exerted equally in all directions. That it must be so is evident from the nature of a fluid, whose particles are perfectly movable among one another, so that any particle could never be at rest unless when equally pressed on all sides. The first inference from this is, that the pressure of a liquid on any surface is
proportional to the area of that surface. Suppose the box B to be filled with water, and to have a number of openings of the same size, as a and b, with pistons or plugs exactly fitting them; and, for greater simplicity, suppose the water to be without weight, so that we may consider merely pressure arising from the forcing down of the plug. If the piston at a be pressed in with a certain force—say, equal to a weight of one pound-this pressure will be transmitted to all parts of the vessel (because the particles of the liquid could not be at rest unless there was an equal pressure throughout), and thus to the piston at b. And since the piston is of the same size as that at a, the pressure on it is the same as the reaction of the water on the piston at a; in other words, there is a pressure of one pound on it. The pressure on the two, then, is two pounds, and both being equal in size, the area of the two is twice that of one of them. If there were a large piston, c-say, four times as large as a-the pressure on it would be four pounds. If a piston, one square inch in area, were pressed into a vessel full of water with a force of one pound, and if the area of one of the sides of the vessel were one foot, then the pressure of the water on that side would be 144 pounds.
On this principle there has been constructed a very useful and powerful machine, named the Hydraulic Press, which is also called Bramah's Press, from the name of the inventor. The figure (19) shews the essential parts of the machine. H is a force-pump by which water is A forced from the tank T, through the tube G, to F, the cavity of a strong B cylinder, E. D is a piston which passes water-tight through the top of E, and is pressed upward by the pressure communicated to the water by the piston of the force-pump H. On the top of D is a plate, on which are placed the articles to be pressed, C; and the rising of the plate, caused by D being forced upward by the water, presses these against another plate, AA. It is very easy to calculate the pressure communicated to D; for, according to the law stated above, the pressure caused by the piston of H is to the pressure on D as the area of the piston is to the area of the end of D. If the area of the end of D were 1000 times greater than that of the piston of H, and the piston of H were pressed down with a force of 500 pounds,
the pressure on D, and through D on the articles between the two plates of the press, would be 500,000 pounds, or above 200 tons.
The pressure caused by a piston being forced into a vessel filled with water, may also be caused by the weight of water itself; for whether the piston at a (fig. 18) be forced in with a pressure of one pound, or one poundweight of water stand in the tube, the pressure in both cases is the same. In this way a very strong cask may be burst by a few ounces of water. In fig. 20, a is a cask filled with water, and b is a very narrow tube inserted in the top of the cask. If the tube hold only half a pound of water, and the bore of the tube be one-fortieth of a square inch, the pressure of the water in the tube will cause a pressure, transmitted by the water in the cask, of half a pound on every one-fortieth of an inch of the inner surface of the cask-that is, of nearly 3000 pounds on every square foot a pressure which no ordinary cask could bear.
This bursting of the cask is an illustration, on a small scale, of the simple means by which the operations of nature are a effected. For example, the water poured into a crevice in a rock by successive falls of rain will ultimately rise to such a height as to cause a pressure sufficient to burst asunder from the mass a large portion of the rock.
2. In a liquid mass, there is a pressure increasing in intensity with the perpendicular depth. The truth of this will be at once clear if we suppose a mass of liquid divided into thin horizontal layers. The upper layer must be supported on the second, and the pressure on the second layer is the weight of the first. And, since the second layer must be supported by the third, the pressure on the third layer is the weight of the two above it; and so on to any depth, the pressure at any depth always being the weight of the water above, so that it must always be proportional to the perpendicular depth. And this is the case whatever may be the shape or width of the vessel. Every one must have noticed with how much greater force water rushes from a deep vessel when the opening is near the bottom, than when it is near the top, or when the vessel is nearly empty this difference is caused by the difference in the pressure of the water above.
3. The free surface of a liquid mass in equilibrium is a perfect level. Since the particles of a liquid are freely movable among one another, it follows that if the liquid were heaped up at any particular part it would always slide down again till the surface was level. There is a case, however, in which the fact is not perfectly clear at first sight-namely, when the vessel consists of different parts communicating with each other. A common tea-pot will afford a convenient illustration. The water stands
in the spout at the same height as in the pot itself; and it must do so, for if the water at B, for instance, were lower than at A, then the pressure under A would be greater than under B, since (§ 2) the pressure is proportional to the depth of the liquid; the water, therefore, could not be at rest, B but a flow must take place into the spout, which would continue till the pressure in it was the same as in the pot, that is, till the water stood at the same level.
This principle is applied to the introduction of water into houses in towns. As fluids always rise to a level, Fig. 21. no matter what distance the water may be conveyed by pipes, it will rise to the height of the source from which it is brought. 4. A solid body, immersed in a liquid, experiences a pressure equal to the weight of the liquid which it displaces, and this pressure acts vertically upwards through the centre of gravity of the liquid displaced. Let AB be a solid body immersed in water; it is evident that AB occupies the place of a quantity of water equal in volume to itself. Now, suppose AB not yet placed in the water, and AB, as seen in the figure, to be the water about to be displaced; this part of the liquid is supported by the pressure of the rest around. The pressure on the sides has no effect, because it is equal all round, and may therefore be disregarded: it is the pressure from below that properly supports the mass. And since this mass of water has a certain weight which acts at its centre of gravity g, the upward pressure keeping it in its place must be equal to that weight, and must act through its centre of gravity. Suppose, now, the solid to be substituted for the water, it must experience exactly the same pressure as acted on the water; that is, the solid AB is acted on by a pressure equal to the weight of the water it displaces, and acting vertically upward through the centre of gravity of the water displaced.
It is an obvious corollary from this, that if a solid be weighed in a liquid, it will be lighter (than its true weight) by the weight of a quantity of the liquid equal in volume to the solid.
This truth was first discovered by the ancient mathematician, Archimedes, and by means of it he was able to discover how much alloy the goldsmith, whom the king of Syracuse had commissioned to make a crown of pure gold, had fraudulently mixed with the metal. It is said that, one day when floating in his bath, it occurred to him that what was supporting his body was that which would support the water displaced by it; and he thought he could, by means of this principle, discover whether
the crown was of pure gold. He is reported to have been so overjoyed at the discovery, that he forgot to dress himself, and rushed through the streets, crying: 'I have found it! I have found it!' To test the crown, he first found the absolute weight of a piece of pure gold, and then its weight when immersed in water. He treated the crown in like manner, and found that it displaced more water in proportion to its weight than the piece of pure gold, which proved that the metal had been mixed with something lighter.
5. A floating body displaces its own weight of the liquid. We have seen that, when a body is immersed in water, a pressure equal to the weight of the water displaced acts upon it, pushing it upward, while its own weight tends to make it sink. If, then, these two pressures are equal, the body will rest in any part of the liquid; if the weight of the body is greater than the weight of an equal volume of water, the body will sink to the bottom; while, if the weight of a quantity of water, equal in volume to the body, is greater than the weight of the body, it will be forced upward to the surface. When this last takes place, the body is said to float. And, of course, part of it must rise above the surface, because, if it were to rest with its surface exactly level with the surface of the liquid, it might have rested at any part of the liquid; but it is
supposed to have been forced upward to the surface. The question, then, comes to be, How much of the body will rise above the surface? or, which is the same thing, How much of it will be in the water? We have seen that a body, immersed in a liquid, remains at rest when the weight of a quantity of the liquid equal in volume to itself, is equal to its own weight; and from this it is clear that a floating body (as AB) will be at rest, when the weight of the water displaced by the submerged part B is exactly equal to the weight of the body. Thus the heaviest bodies can be made to float on a liquid, if only they can be so arranged as to displace a quantity of the liquid of weight greater than their own. A piece of iron sinks in water, but ships can be made of iron, because they are hollow, and displace a quantity of water of greater weight than their own.