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IV.-The Inclined Plane. The Inclined Plane is a contrivance by which a heavy weight can be raised to a height, when it would be difficult or impossible to lift it directly. We have seen, in the case of the Mechanical Powers already considered, that no additional power is actually gained by the use of the machine, or is generated by it, but that its use is simply to enable a small force, when exerted over a larger space, or in a more extended manner, to have the same effect as a greater force applied directly: the same principle applies to the inclined plane. A force pushing a weight from A to B (fig. 13) only raises it through the perpendicular height BD. If, then, AB
be twice as long as the height BD, 2
the force necessary to raise any weight from D to B, would, when
pushing the weight from A to B, be P distributed over twice as much space:
but in consequence of this, the force exerted at any moment would be only half as great, leaving out of
account the effect of friction, which Fig. 13.
cannot be taken notice of here. Thus,
a power, P, would balance a weight on the plane twice as great; that is, if the mass W weigh two pounds, the weight of P would require to be only one pound. The reason of this is, that part of the weight of W is supported on the plane, so that the resistance of the plane, and the tension of the string, P, support the weight of W between them, in the way represented in the figure. The weight of W acts perpendicularly downward from its centre of gravity, c, in the direction of cf; now, the part of this weight supported by the plane must act perpendicularly to the plane, therefore along ce; and the part of it
supported by the string, directly opposite to the tension of the string, along cg; and these two are independent of each
other, so that the pro
'c portion of the weight of Fig. 14.
W to be supported by the
string is as ce is to cf. Now, it is known, according to a simple geometrical problem, that cg is
i Centre of gravity is a point in a body such that, if the body be suspended upon that point, it will balance itself in every position.
to cf as BD is to AB; therefore, the force necessary to balance a weight on an inclined plane is proportioned to that weight, as the height of the plane is to its length. The degree of slope in an incline is expressed by saying that there is a rise of one foot in so many feet ; thus, in fig. 14, where AB is ten times BC, the horse is pulling a load up an incline of one in ten. And, according to what has been said, the draught necessary to be exerted by the horse, if the plane be perfectly smooth, is only one-tenth of the weight of the load. More correctly, a tension of one-tenth the weight of the load would keep the cart from rolling backward ; a little more exertion on the part of the horse would be necessary to pull it up.
V.-The Screw. The Screw is an application of the inclined plane. It is simply an inclined plane passing round a cylinder, as can be shewn by a very simple experiment. If a triangular piece of paper, such as abf in fig. 15, be wound round a pencil or ruler, the upper edge of the paper, which is evidently an inclined plane, will form spiral exactly like a screw, except that a screw is formed by a
Fig. 15. spiral ridge. This spiral ridge is called the thread of the screw, and it works in a nut (M, fig. 16), which has a spiral groove to receive the thread, and is turned by means of a lever, L. When the screw is turned once round, it is carried forward in a fixed nut, or it draws forward a movable nut upon it, through the space between two of its threads. The resistance, therefore, has been pulled forward a distance, eg or ga' (fig. 15); but it has actually passed along the inclined plane de, supposing de to be the part that would go round the pencil. Therefore, the resistance overcome by the screw is to the force exerted in turning the screw as de is to ec. A screw, however, seldom acts by itself, being generally turned by a lever, so that, by means of the lever and the screw, a man is often able to overcome a resistance more than a hundred times greater than the force he applies to the lever.
VI.-The Wedge. The Wedge (fig. 17) is a contrivance for separating or overcoming resistances by being forced in between them, and is really a combination
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. Fig. 17.
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.
HYDROSTATICS. 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 statikē, 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
16 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
Fig. 18. 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 á force-pump by which water is A forced from the tank T, through the tube G, to F, the cavity of a strong 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
т 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
Fig. 19. 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 ery 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