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longer ends. And in a combination of wheels and axles, the power will sustain the weight, when the power is to the weight as the product of all the diameters of the axles is to the product of all the diameters of the wheels. And in general, in every machine, the power will sustain the weight, when they are to each other inversely as their velocities. So that what is gained in power is lost in time; and it is impossible to do otherwise.
In estimating the power, the weight of the matter of which the machine is made, and the friction of its parts, must be added or subtracted, according as it is for or against the power. The softer or rougher that bodies are which rub upon each other, the greater, in general, is the friction; if their pressure be increased the friction will be increased in the same ratio: and if their velocity be increased, the friction will also be increased, but not in the same ratio. In general, the increase or diminution of smooth surfaces makes no alteration in the friction while the weight and velocity continue the same. Wood slides more easily on the ground in wet weather than in dry, and more easily than iron in dry weather, but iron more easily than wood in wet weather. A cubic piece of soft wood, eight pounds in weight, moving at the rate of three feet per second on a smooth plane of soft wood, has a friction equal to two thirds of its weight nearly; but moving on a smooth plane of hard wood, its friction is equal to about one sixth of its weight: and hard wood upon hard wood has a friction equal to about one eighth part of its weight. Two thirds of the friction of wood upon wood may be destroyed by
oil or grease properly applied. The naves of wheels have four times less friction, when greased than when wet. When polished steel moves upon polished steel or pewter properly oiled, the friction is about one fourth of its weight; on copper or lead, about one fifth; and on brass, one sixth of its weight.
The friction of ropes moving on machines depends upon their stiffness, tension, grease and state of the weather. All other circumstances being equal, the difficulty of bending a rope, is as the square of its diameter and its tension directly, and inversely as the diameter of the cylinder or pulley, about which it goes. A rope of one inch diameter tended by five pounds and passing over a pulley of three inches diameter requires a weight of one pound to bend it. In the pulley, the friction is directly as the weight, the velocity, and diameter of the axis; and inversely as the diameter of the pulley. A single pulley loaded with 14 pounds at each end of the rope, requires one pound to be added to one end to raise the weight at the other. And a power of one hundred pounds that would sustain a weight of 500 pounds with a combination of pullies of 5 ropes requires an addition of 50 pounds to raise the weight. In the wheel and axle, the friction on the axle is as the weight, the velocity, and diameter of the gudgeon. On the inclined plane, the friction is according to the matter and surface of the rubbing bodies, and the bending of the rope and rub. bing of the pulley, as explained above.
EQUABLE motion is that which is generated in a body by a single impetus; by which the body will pass over equal spaces in equal times. The spaces
therefore must be proportional to the times; that is, if it pass over a mile in an hour, it will pass over two miles in two hours, if the velocity continue the same without variation. The spaces also, that are passed over in a given time, must be proportional to the velocity of the motion; for if with a certain velocity it pass over a mile in an hour, with double that velocity it would pass over two miles in the same time. Hence as the spaces depend on both the time and velocity of the motion, they will be proportional to the rectangle of both. Let S = the space, T= the time, and V=
the velocity; then S=TV. V= and T=
=v. Hence the velocity will be directly as the space and inversely as the time; and the time will be directly as the space and inversely as the velocity. And if the spaces be equal or given, the times and velocities will be in an inverse ratio to each other; that is the less the time is, the greater must be the velocity in order to pass over the given space.
IN BODIES DESCENDING BY GRAVITY, IN VACUO.
ACCELERATED motion is produced by the continual operation of the moving force every instant; generating an additional velocity every moment. And as gravity is such a force as continually exerts its influence on descending bodies, to bring them in right lines towards the center of the earth, their motion must be continually accelerated: and as it acts with equal force on the same body at all times, the increments of velocity produced by its exertion must be equal in equal times; that is the motion of heavy bodies descending freely near the earth's surface, by
the force of gravity, in spaces void of all resistance, must be equably accelerated. Secondly, the increments of velocity generated in every indefinitely small portion of time, being equal to one another, the aggregate or sum of them, at the end of the fall must bear a constant proportion to the time in which they were produced: that is, the last acquired velocity is always proportional to the time of the fall. Thirdly, the spaces passed over by descending bodies are as the squares of the times, or as the squares of the last acquired velocities. For the spaces depend both on the times and velocities of the body passing over them; but these times and velocities are proportional to each other; so that one may be substituted for the other. Therefore the spaces will be as the squares of the times, or squares of the last acquired velocities. Hence the times or velocities will be as the square roots of the spaces. Fourthly as the velocity begins from nothing and gradually increases to the end of the fall, it is evident that if the body had moved with the last acquired velocity during the whole time of the fall, it would have passed over a space double to what it passed over in its descent, or it would have passed over the same space with the mean velocity it had acquired in the middle point of the descent. Fifthly, it is found by accurate experiments that a heavy body falls by the force of gravity, in spaces void of resistance, 16.15 feet in the first second of time. Hence by such a fall it would acquire a uniform velocity of 32.3 feet per second. Sixthly, as gravity would generate in the descending body an uniform velocity in the first second, which would in the same time carry it over twice 16. 15 feet in another second, it is evident, that if gravity ceased to act, the body would have passed over thrice 16.15 feet at the end of two seconds of
time; but as it continues to act with the same force which it exerted in the first second, the space passed over must be four times 16.15 feet at the end of this second. It has then acquired an uniform velocity, that would carry it over eight times the space of 16.15 feet in the third second, if gravity ceased to act; but during the third second gravity continues to act as formerly and will carry the body over another 16.15 feet; so that at the end of the third second, it has descended through nine times 16.15 feet. But in the two first of these seconds, it had passed over four of these spaces, and in the third five of them. Hence the spaces descended through in the separate second of time, will be as the odd numbers, 184.108.40.206.9, &c. And hence as the square of 1 is 1. and the sum of 1 and 3, viz. the sum of the spaces passed over at the end of two seconds is 4, the square of two seconds; and the sum of 4 and 5, the spaces passed over at the end of three seconds, is 9, the square of three seconds; and the sum of 9 and 7, the spaces passed over at the end of four seconds, is 16, the square of four seconds, &c. It is evident that the spaces descended through at the end of any time must be as the square of the time, as we have seen before.
Now as these odd numbers grow large, they approach nearer to an equality, and consequently as they represent the spaces passed over in each separate portion of time, the motion approaches nearer to an equable motion, the longer it is continued.
Bodies projected perpendicularly upwards are retar. ded by gravity in the same manner as they are accelerated by gravity in their descent. Hence they spend the same time in rising to a given height, as in falling from it. The heights to which they will rise, when projected with any given velocities will be as the squares