< 2 x2=16: wherefore, if the weight be one lb., it will De sustained by a power of one oz. avoirdupois.

31. When there are any number of movable pulleys on one block, and as many on a fixed block, the pulleys are called Sheeves, and the system is called a Muffle; and the weight is to the power inversely as one is to twice the number of movable pulleys in the system, or

the weight to be raised

twice the number of mov. pulleys

=

the power.

Ex. In a muffle where each block has 4 sheeves, one block being fixed and the other movable, a weight of 112 lbs. is to be raised; how great must be the power?

If a power of 236 lbs. is to be applied to a tackle con nected with two blocks of pulleys, one fixed, consisting of 6, and another movable, of 5 pulleys; what weight can be raised (Here the rule above must be reversed.)

Therefore 236 x 10 = 2360 lbs., the weight.

REMARK. In all the above cases of the pulley, the strings, cords, or ropes, are supposed to act parallel to each other; when this is not the case, the relation of power and weight may be found by applying the principle of the parallelogram of forces; thus, draw ab in the direction of the rower's action and of that length, taken from a scale of equal parts, which expresses the quantity of that power; next, draw bd a perpendicular to the horizon, and from a draw ad parallel to bc, the direction of the string, which is fastened at c: then the power is to

a

the weight, as ba is to bd; and the strain on the hook at c, is as ad to db,-these lines being all measured on the same scale of equal parts.

It may be further observed, that the pulley is a species. of lever of the second kind; where the point at which the string is fastened may be called the fulcrum; the axis of the pulley the place of the weight, and the place of the power the other end of the lever; or, the diameter of the pulley may be reckoned the length of the lever, the weight being in the middle.

THE INCLINED PLANE.

32. WHEN a power acts on a body, on an inclined plane, so as to keep that body at rest; then the weight, the power, and the pressure on the plane, will be as the length, the height, and

the base of the plane, when the power acts parallel to the plane; that is,

The weight

AC,

These properties give rise to the following rules :

weight x height of plane.

power

=

length of plane

weight

power x length of plane

=

height of plane

weight base of plane

length of plane

33. The force with which a body endeavours to descend down an inclined plane, is as the height of the plane. When the power does not act parallel to the plane, then from the angle C of the plane, draw a line perpendicular to the direction of the power's action; then the weight, the power, and the pressure on the plane, will be as AC, CB, AB.

B

When the line of direction of the power is parallel to the plane, the power is least.

34. If two bodies, on two inclined planes, sustain each other, by means of a string over a pulley, their weights will be inversely as the lengths of the planes.

35. In the exercises on inclined planes, it is often necessary to find the length of the base, and height, or length of the plane. Any two of these being given, the third may be found and this is done on the principle stated in Geometry, that the hypotenuse of a right-angled triangle (the length of the plane) is equal to the base+height".

2

Ex. The height of an inclined plane is 20 feet, and its length 100; what is the pressure on the plane of a weight of 1000 lbs.?-Here we must first ascertain the base, the base of the plane; and from

(1002—20o) = 97.98 =

what has been said above, 100: 1000 :: 97-98.979-8 the pressure upon the plane; also 100: 20 :: 1000 200, the power necessary to keep the body from rolling down the plane.

If a wagon of 3 cwt. on an inclined railway of 10 feet to the 100, be sustained by another on an opposite railway of 10 feet to 90 of an incline; what is the weight of the second wagon?-Here 100: 90 :: 3 : 2·7 cwt. the weight of

the second wagon.

=

36. The space which a body describes upon an, inclined plane, when descending on the plane by the force of gravity, is to the space which it would fall freely in the same time, as the height is to the length of the plane; and the spaces being the same, the times will be inversely in this proportion.

Ex. If a body roll down an inclined plane 320 feet long, and 26 feet in height; what space will it pass down the plane in one second, by the force of gravity alone?

320: 26: 16: 1.3 foot = the answer.

This subject, as connected with railways, will be resumed when we come to treat of friction and railways.

THE WEDGE.

37. THE wedge is a triangular prism, formed either of wood or metal, whose great use is to split or raise timber, stones, &c.

The circumstances in which it is applied are such that it is not easy to devise a general rule to determine the amount of its action. The wedge has a great advantage over all the other mechanical powers, in consequence of the way in which the power is applied to it, namely, by per cussion, or a stroke, so that by the blow of a hammer, almost any constant pressure may be overcome.

THE SCREW.

38. THE screw is a kind of continued inclined plane, being an inclined plane rolled about a cylinder-the height of the plane being the distance between the centres of two threads, and its length the circumference; hence,

the rule to find the power of a screw pressing either upwards or downwards, is as the distance between two threads of the screw is to the circumference where the power is applied thus, if the distance of the centres of two threads of the screw be of an inch, and the radius of the handspike attached to the screw be 24 inches; the circumference of the screw will be 150 inches, nearly: therefore,

: 150 :: 1: 603; and if the power applied be 150 Ibs., the force of the screw will therefore be 6031⁄2 × 150 = 90480 lbs.

39. REMARKS ON THE MECHANICAL POWERS.-The mechanical powers may be variously modified and applied, but still they form the elements of all machinery. In our calculations of their effects, we have not made allowance for friction, or the resistance arising from one body rubbing against anotherr-a subject which will be discussed hereafter. The justice of the remark made before, will now be seen to hold generally, that of the two-velocity and powerwhatever we gain in the one, we lose in the other; or, as power and weight are opposed to each other, there will always be an equilibrium between them, when the power X its velocity the weight x its velocity, that is, when the momentum of the one is equal to the momentum of the other.

=

All the advantage that we can obtain from the mechanical powers, or their combinations, is to raise great weights, or overcome great resistances, and this must be done at the expense of time; or, to generate rapid velocities, as in turning-lathes, or cotton-spinning machinery, and this is done at the expense of power.

MECHANICAL CENTRES.

1. THESE are the centres of gravity, oscillation, percussion, and gyration.

THE CENTRE OF GRAVITY.

2. THERE is a certain point in every body, or system of bodies connected together; which point, if suspended, the

body or system of bodies will remain at rest when acted upon by the force of gravity alone;-this point is called the Centre of Gravity. If a body or system of bodies be suspended by any other point than the centre of gravity, such body or system of bodies will move round that point, until the centre of gravity be in a vertical line with the point of suspension. If a body be sustained from falling by two forces, the lines of direction in which these two forces act, will meet in the centre of gravity of the body, or, in the vertical line which passes through it.

3. It is often useful in calculation to consider the whole weight of a body as placed in its centre of gravity, but it' is to be remembered, that gravity and weight do not signify the same thing-gravity is the force by means of which bodies, if left to themselves, fall to the earth in directions perpendicular to the earth's surface; weight, on the other hand, is the resistance or force which must be exerted, to prevent a given body from obeying the law of gravity.

4. To find the centre of gravity of any plane figure, mechanically Suspend the figure by any point near its edge, and mark the direction of a plumb-line hung from that point, then suspend it from some other point, and mark the direction of the plumb-line in like manner. The centre of gravity of the figure will be in that point where the marks of the plumb-line cross each other. For instance, if we wish to find the centre of gravity of the arch of a bridge, we draw the plan upon paper to a certain scale, cut out the figure, and proceed with it as above directed; and by means of the plumb-line from the points of suspension, its centre of gravity will be found; whence, by measuring the relative position of this centre in the plan by the scale, we may determine by comparison its position in the structure itself.

5. We can find the centre of gravity of many figures by calculation.

6. The centre of gravity of a line, parallelogram, prism, cylinder, circle, circumference of a circle, sphere, and regular polygon, is the geometrical centre of these figures respectively.

7. To find the centre of gravity of a triangle-draw a line from any angle to the middle of the opposite side, then