THE PULLEY.

40. Definition of PULLEY.

A PULLEY is a small wheel moveable about an axis through its centre, and having a groove of uniform depth along its outer edge to admit a rope or flexible chain. The ends of the axis are fixed in a frame called the Block.

The Pulley is said to be fixed, or moveable, according as the Block is fixed, or moveable. The Power acts by the rope, or chain, or string, which works in the groove; and the Weight is fixed to the Block.

In the following Propositions the groove of the Pulley is supposed perfectly smooth; and the weight of the machine is not taken into account.

41. PROP. XI. In the single moveable pulley, where the strings are parallel, there is an equilibrium when the Power is to the Weight as 1 to 2*.

The annexed diagram represents the case here supposed. A power, P, acts by means of a string PADBR, Ry which passes under the moveable pulley and is made fast at R. A weight, W, is fixed to the block by a string, and the line of its action is CW, C being the centre of the section of the pulley made by a plane passing through PA B and RB. By the supposition, PA, RB, and CW, are parallel; and they are in one plane. Also PA, RB, are tangents to the circle at A and B; therefore, if AC, BC, be joined, PAC= a right angle RBC, and .. ACB is a straight line, and CW is at right angles to AB.

Now, the whole machine being at rest under the action of these forces, we may consider it as a straight lever AB kept at rest round B, as a fulcrum, by the power Pacting perpendicularly upwards at A, and the weight W acting

• More correctly thus:-"when there is equilibrium, the Power is to the Weight &c."

perpendicularly downwards at C; but in this case, and therefore in the case first supposed, the forces are inversely as their distances from the fulcrum, that is,

P: W: BC: AB 1: 2.

42. PROP. XII. In a system in which the same string passes round any number of pulleys, and the parts of it between the pulleys are parallel, there is an equilibrium when Power (P): Weight (W) :: 1: the number of strings at the lower block*.

The annexed diagram represents such a case as is here supposed. The system consists of two blocks, having each two pulleys, the upper block being fixed, and the lower one moveable. The string, by which Pacts passes round each of the pulleys, as shewn in the figure the several portions of it are parallel to each other, and to the line AW in which the Weight (W) acts.

M

YP

Since the same string continuously passes round all the pulleys, its tension must be everywhere the same, otherwise motion will ensue, which is contrary to the supposition. But for the outer portion of the string, to which the power is immediately applied, this tension is P; therefore it is P throughout; that is, each string at the lower block exerts a force P in opposition to W. And the forces are all parallel, therefore, since they balance, Wis equal to the sum of those which act in the opposite direction, that is, is equal to P multiplied by the number of strings at the lower block, or P: `W:: 1: No. of strings at the lower block, whatever that number may be.

43. PROP. XIII. In a system in which each pulley hangs by a separate string, and the strings are parallel,

More correctly thus:-"when there is equilibrium, the Power is to the Weight &c."

there is an equilibrium when P: W:: 1 that power of 2 whose index is the number of moveable pulleys*.

In this system the strings are fixed at F, G, H, &c. and pass round the moveable pulleys A, B, C, &c. respectively, as in the figure, to the last of which the power P is applied. The strings are all parallel, and in the direction in which the weight W acts.

Now, C being a single moveable pulley with FGH parallel strings, when there is equilibrium, by Prop. XI., the pressure downwards at C-2P; therefore the Tension of the string from Cround the pulley B=2P.

Hence, the weight supported at the moveable pulley B = 2× (2P) = 22×P= tension of string which passes round the pulley A.

So, weight supported at moveable pulley A = 2×(23×P) = 23×P.

WV

Therefore, when there is equilibrium on the system of three moveable pulleys, as here represented,

W = 23×P.

AP

And so on, by the same mode of reasoning, if n be any number of moveable pulleys, it will appear, that, when there is equilibrium, W=2"×P;

44. Definition of INCLINED PLANE.

An Inclined Plane, in Mechanics, means a plane inclined to the horizon, that is, neither horizontal, nor vertical.

The plane is here supposed to be perfectly smooth, and rigid, and capable of counteracting, and entirely destroying, the effect of any force which acts upon it in a direction perpendicular to its surface.

When, therefore, a body is sustained on an inclined plane, by a Power directly applied to it, the case is that of a body kept at rest

* More correctly thus:-"when there is equilibrium, P: W &c."

P

by three forces, its own Weight acting vertically, the power applied, and the resistance of the plane in a direction at right angles to its surface. The Power must also obviously act in the same plane as the other two forces, otherwise motion would ensue.

It is evident, that the Inclined Plane will bear, or take off, a portion of the Weight, how much is the question; that is, we are required to find the ratio of P to W, when there is equilibrium.

45. PROP. XIV. The weight (W) being on an Inclined Plane, and the force (P) acting parallel to the plane, there is an equilibrium when P: W:: the height of the plane : its length*.

[The 'length' of the inclined plane is the section of it made by the plane in which the forces act-the 'height' of the plane is the perpendicular let fall from the highest point in it to meet the horizontal plane through the lowest point—and the 'base' is the distance from the lowest point to this perpendicular. Thus, in the annexed diagram, AB is the 'length' of the plane, BC its 'height', and AC its 'base'.]

B

Let AB be the length of the Inclined Plane; AC its horizontal base; and BC, perpendicular to AC, its height. Let the Weight (W) be supported on the plane at D by the Power (P) acting in the direction DB parallel to the plane; then, when there is equilibrium,

P: W:: BC : AB.

F

From D draw DE at right angles to AB, meeting the base in E; and from E draw EF vertical, or at right angles to AE, meeting AB in F.

Then in the triangles EFD and ABC, FE, BC are parallel, angle DFE= angle ABC;

and angle FDE= a right angle = angle BCA,

.. angle DEF-angle CAB, and .. the triangles are equiangular and similar. Hence DF: FE :: BC: AB.

• More correctly thus:-" when there is equilibrium, P : W &c.”

Now the body at D is kept at rest by three forces, the weight Wacting vertically, the reaction of the plane acting at right angles, and the power P acting parallel, to the plane. And these forces are respectively parallel to the sides of the triangle DEF, therefore those sides will represent them in magnitude as well as in direction (by the converse of Prop. IX.). Hence

P: W: DF: FE,

:: BC : AB, (by what has been proved.)

46. Definition of VELOCITY.

By the VELOCITY of a body in motion is meant the Degree of Swiftness, or Speed, with which the body is moving. And this "degree of swiftness" is described, or measured, by stating how long the line is that the body moves through, with uniform swiftness, in some given portion of time.

Thus a clear notion would be conveyed of the Velocity of a coach, if it were said to be nine miles in an hour; the space moved over by the coach being nine miles, and an hour being the portion of time during which the motion took place. If only the space that is moved through were mentioned, and nothing were said about the time of describing that space,-or if the time only were given,— it is evident that no idea could be formed of the degree of swiftness, that is, of the Velocity, of the coach.

The Velocity of a body is measured, for the most part, in mathematical investigations, by the number of feet passed over by the ⚫ body, moving uniformly, in a second of time.

COR. Since the quicker a body moves the more space it will pass over in a given time, it will follow from the observations just made, that The Velocities of two bodies which move during any (THE SAME) time, are in the ratio of the spaces which the bodies respectively describe each with uniform swiftness—in that time.

47. PROP. XV. Assuming that the arcs which subtend equal angles at the centres of two circles are as the radii of the circles, to shew that, if P and W balance each other on the Wheel and Axle, and the whole be put in motion, P: W:: W's velocity: P's velocity.

L. C. C.

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