if there be feveral folds thereof round the axis, you must measure to the middle of the outfide rope to obtain the radius of the axis; for the diftance of the weight is increased juft fo much by the coiling of the rope round the axis.

You have seen that the velocity of a weight, or any other force applied to the axis of a wheel, will increase as the circumference of the axis increases. Thus a weight, or any other force acting upon the axis B, will have a greater velocity than one which acts upon the axis A; if the force was applied to T, it's velocity would be ftill further increased. Now the fufee of a watch is an axis of the shape of a cone, and the spring by means of a chain acts upon the fmalleft part of this axis when the watch is wound up; as the watch goes down, the chain is applied to different parts of this cone, and to the thickeft part of all when the watch is juft down. Therefore as the elafticity or abfolute force of the fpring decreases, the velocity with which it acts to move the watch increases, by being applied to a larger part of the axis in proportion; and confequently if the fufee be properly regulated, the momentum produced in the wheels and the hands of the watch, will be always the fame. In other words, the motion of a watch is rendered uniform by the fufee, though the force of the fpring is not always the fame, but is greatest when the watch is wound up, and keeps decreasing as the watch goes down.

OF THE PULLEY.

The pulley is a fmall wheel turning about it's axis, with a drawing rope paffing over it; the fmall wheel is commonly called a beeve, and is fo fixed in a box or block, as to be moveable round a center pin paffing through it.

VOL. III.

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Pullies

Pullies are of two kinds: 1ft, Fixed, which do not move out of their place. 2d, Moveable, which rife and fall with the weight.

A rope going round one or more pullies to raife a weight, is called the running rope of the fixed pulley; this kind only turns on it's axis, but does not move out of it's place, fee B, fig. 4, pl. 3; or V, fig. 3, pl. 5. It changes the direction of the power, but gives it no mechanical advantage; fo that you can raife no greater weight by means of this pulley than you can raife without it by your natural strength.

Two equal weights fufpended to the ends of a ftring that goes over a fixed pulley, will ballance each other, for they are equally ftretched by the weights; and if either of them be pulled down through any given space, the other will rife through an equal space in the fame time; and confequently as the velocity and the weights are equal, they muft ballance.

Though this pulley gives no mechanical advantage, it is a fource of great conveniency, as it takes off the neceffity a man would otherwife be under of afcending along with the weight, and thus leffens his labour; befides having this further convenience, that by means thereof the joint ftrength of feveral perfons may be made use of to raise one and the fame weight.

OF THE MOVEABLE PULLEY,

This kind C, fig. 4, pl. 3, or T, fig. 3, pl. 5, rifes or falls along with the weight, and adds to the momentum of the power.

Some writers have thought that the nature and effects of the pulley might be beft understood by confidering a fixed pulley as a lever of the first kind, and a movcable pulley as one of the fecond.

But

But this mode feems only calculated to introduce the parade of demonftration, for the pulley cannot with propriety, be called a lever; for when any power fuftains a weight by means of a fyftem of pullies, that power will fuftain the fame weight if the pullies be removed, and the ropes be brought over the axis on which the pullies turned. Now in this cafe, no one I believe would fay, that these axes fhould be confidered as levers; if it was required to raise a weight, there would be a very great refiftance arifing from the friction of the ropes on the axes, and it is merely to avoid this refiftance that pullies are used, because they turn upon their axis with very little friction.*

The best and most natural method of computing the proportion of any power, to the weight it fuftains by means of a system of pullies, or of explaining their effects, is by confidering, that every moveable pulley bangs by two ropes equally fretched, and which must confequently bear equal parts of the weight; and therefore when one and the fame rope goes round feveral fixed and moveable pullies, as all the parts on each fide are equally tretched, the whole weight must be equally di vided among all the ropes by which the moveable pullies are fufpended.

Hence if the power which acts on one rope be equal to the weight divided by the number of ropes, that power muft fuftain the weight; in other words, the power and weight ballance each other, when the power is in proportion to the weight, as one is to the whole number of cords. Upon this principle the proportion of the power, to the weight it fuftains by means of a fyftem of pullies, may be computed in a manner fo eafy and

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* Hamilton's Four Introductory Lectures.

natural

which a separate ftring goes round each pulley; one end thereof is fixed, while the other is joined to the moveable pulley next above it. There will be an equilibrium, when the power is to the weight as unity to that power of two, of which the index is the number of moveable pullies. See F, fig. 4. pl. 3.

The weight is faftened to the lower pulley, The pullies do not here, as in the preceding fyltems, rife together in one block with the weight, but act upon one another, fo that each pulley doubles the power.

A power of one ounce will be in equilibrio with two ounces at A, with four at B, with eight. at C, and with fixteen at D; and the velocity of the power will be fixteen times that of the weight. Though this fyftem be very powerful, yet as it raifes weights very flowly, and takes up a confiderable space, it is very seldom used.

OF THE INCLINED PLANE.

The inclined plane is generally reckoned among the mechanic powers, being ufed with advantage in raifing weights, by diminishing the weights laid thereon.

If hogfheads, or pipes of wine, &c. are to be let down into a cellar, or brought up out of it, a plank is laid along the ftairs, which in that cafe is an inclined plane, the only mechanical power that can be readily applied. So alfo, in making refervoirs for water, in gardening, in raifing fortifications, &c. where carts cannot come, inclined planes made of wood ferve for the wheelbarrows to run on, to remove the earth from a lower to an higher place.

I have already fo far explained to you the properties of the inclined plane, that need do

little more, in this place, than fhew you, that the lefs the plane is inclined to the horizon, the cafier a body may be rolled or forced up it's furface.

If it were required to raise a heavy body to the height CB of the plane, fig. 6, pl. 4, and perpendicular to the horizon, it is evident you must employ a power equal to that of the weight, and it would even then be very inconvenient; but if an inclined plane, A B, be elevated to the height B, where the weight is to be raised, a lefs power than the weight will ferve the purpose.

The weight is always moft eafily either drawn or puffed in a line W w M, parallel to the plane, and paffing through the center of the weight; for if one end of the line be fixed at W, and the other end inclined towards B, the body would be drawn against the plane, and the power must be increased in proportion to the greater difficulty of the traction: but if the line be carried above M, the power must be alfo increased; but here only in proportion as it endeavours to lift the body off the plane.

As the power acts to the greatest advantage when the line of traction is parallel to the inclined plane, what I fhall mention concerning this plane will principally apply to the line when in this fituation. When fo fituated, there will be an equilibrium, when the power is to the weight as the height of the plane is to it's bafe. In other words, the mechanical advantage is in proportion as the length of the plane exceeds it's height.

Thus, if a weight of four ounces be laid on the plane RS, fig. 2, pl. 5, whofe length is to it's perpendicular height as 2 to 1, it will be counterballanced by a weight of two ounces, drawing the other in a direction parallel to the plane, by means of a pulley properly fixed at the end thereof. If the length of the plane were to it's height as 4 to 1, it

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