SECTION V.

MECHANICAL POWERS, CIRCULAR MOTION, &c.

The Mechanical Powers are the known elements of machinery. They are three in number, with some diversity of application. Strictly speaking, they are not poWERS, or sources of power; they simply convey applied force, and diffuse or concentrate it. In treating of them, the term weight, or resistance, is understood to be the force to be overcome, and the term power, the force applied to overcome or balance it. It is also to be understood that the deductions or conclusions arrived at are theoretically true; that is, that they are true upon the supposition that the whole power employed is expended to the end under consideration — that no friction or weight of machinery aids, or is to be overcome.

Lemma.

THE LEVER.

The power multiplied by its distance from the fulcrum equals the weight multiplied by its distance from the same point; and as the distance between the power and fulcrum is to the distance between the weight and fulcrum, so is the effect to the power.

Consequently, if we divide the weight by the power, we obtain a quotient equal the length of the longest arm of the lever, the length of the shortest arm being 1. And if we multiply. the weight by its leverage, and divide the product by the power, we obtain a leverage for the power that will enable it to equipoise the weight. And if we multiply the power and its leverage together, and divide the product by the weight, we obtain for the weight the same result. So, too, if we divide the lever by the quotient obtained by dividing the weight by the power, to which quotient we have added 1, we obtain the relative position of the fulcrum, or the distance it must occupy from the opposing force. And, again, if we multiply the opposing force by its leverage, and divide the product by the leverage pertaining to the power, we obtain the requisite power to counterbalance the resistance.

Example. — A weight of 1200 lbs., suspended 15 inches from the fulcrum, is to be raised by a power of 80 lbs. ; at what distance from the fulcrum on the long arm of the lever must the power be applied, to accomplish that end?

80 1200 1.25 18 feet. Ans.

EXAMPLE. The lever is 20 feet long, the opposing force 1200 lbs., and the available force 80 lbs. ; at what distance from the former force must the fulcrum be placed, that the two forces may equipoise each other?

1200 80 = 15, and 2015+ 1

EXAMPLE.

14 foot. Ans.

The longer arm of the lever is 183 feet, the shorter arm 11 foot, and the weight to be raised is 1200 lbs. ; what power must be applied to raise it?

18.75 1.25 :: 1200 = 80 lbs. Ans.

EXAMPLE. A man, with a lever 5 feet in length, raised a weight of 2500 lbs. suspended across the lever 9 inches from the further end, which rested on a support; what force did the man exert?

EXAMPLE.- A beam, 20 feet in length, supported at both ends and not elsewhere, bears a weight of 6000 lbs. placed 6 feet from one end; what is the pressure on each support?

2014: 6000: 4200 lbs. on the support nearest the weight. 20: 6:: 6000: 1800 lbs. on the support furthest from the w't. S

WHEEL AND AXLE.

Ans.

The WHEEL and AXLE is a revolving lever. It partakes, in all respects, of the same principles as the preceding. The radius of the wheel is the longer arm of the lever, and the radius of the axle, the shorter. The fulcrum is the point of impact between them at the circumference of the axle.

EXAMPLE. The radius of the wheel is 2 feet, the radius of the axle is 9 inches, and the weight to be raised is 500 lbs.; the weight is attached to a rope wound round the axle ; what power must be applied to the periphery of the wheel to raise it? 2.5.75 500 150 lbs. Ans.

EXAMPLE.

- The diameter of the wheel is 5 feet, the diameter of the axle or barrel, 1 foot, and the power is 150 lbs.; what weight may be raised?

EXAMPLE.

1.55 150 500 lbs. Ans.

and

The power is 150 lbs., the resistance is 500 lbs., the barrel has 9 inches radius; required the diameter of the wheel that will enable the power to equipoise the weight. 150 500 :: 9: 30 in. radius, and 30 X 260 ÷ 12

EXAMPLE.

=

5 feet. Ans.

-The length of the winch (crank) of a crane is 15

inches, the radius of the barrel around which the lifting chain coils is 3 inches, the pinion has 8 teeth, and the wheel 68; required the weight that a force of 30 lbs. applied to the winch will raise.

688 =

8 (8 to 1) velocity of pinion to wheel, and 15 X 8.53 42.5 lbs. exertive force, or force to 1 of applied power gained at the expense of space, and 42.5 lbs..X 30 lbs. (applied power):

-

= 1275 lbs. effective power. Ans. EXAMPLE. The exertive force, or effect to power, of a crane, is to be as 42 to 1, the radius of the wheel to that of the pinion as 8 to 1, and the throw of the winch- its length - the radius of the circle which it describes is to be 14 foot; what must be the diameter

of the barrel?

8.5 X 1.25 10.625

42.5 = .25 X 2 = .5 ft. or 6 in.

Ans. NOTE. By additional wheels and pinions, as in the system of pulleys or block and tackle, which see, the exective force of a crane may be increased to almost any conceivable extent; but always, as with the block and tackle, and as shown in the above exam. ple, at a relative expense to space.

A

THE PULLEY.

W

P

B

A single pulley, fixed and turning on its own axis, affords no mechanical advantage. It serves but to change the direction of the power.

In the common system of pulleys, or block and tackle, the advantage is as the number of ropes engaged in supporting the lower or rising block, to 1 of applied force.

RULE. —1. Divide the given weight by the number of cords leading to, from, or attached to, the lower block, and the quotient is the requisite power to produce an equilibrium.

RULE.-2. Multiply the given power by the number of cords leading to, from, or attached to the lower block, and the product is the weight that may be raised.

RULE.-3. Divide the weight to be raised by the power to be applied, and the quotient is the requisite number of cords that must connect with the lower block.

EXAMPLE. The lower, running, or rising block has 5 sheaves or

pulleys; the fixed or stationary has 4; and the weight to be raised is 2250 lbs. What force must be applied to raise it? Necessarily the end of the rope is attached to the lower block, therefore 9 ropes are attached to or connected with it; hence

22509 = = 250 lbs. Ans.

NOTE. In the Spanish burton, having two movable pulleys and two separate ropes, the effect is to the power as 5 to 1. In a system of 4 movable pulleys and 4. separate ropes, it is as 16 to 1. And in a system having 4 movable and 4 fixed pulleys, and 4 separate ropes, it is as 81 to 1.

INCLINED PLANE.

Lemma. The product of the length of the plane and power is equal to the product of the height of the plane and weight.

The velocity, therefore, or force, or momentum, with which a body descends an inclined plane, impelled by its own gravity, is to that with which the same body would descend perpendicularly through space, as the height of the plane to its length, or as the size of its angle of inclination to radius. And the space the body describes upon the plane, in any given time, compared with that which it would describe falling freely, in the same time, is as its velocity upon the plane to that of perpendicular descent. And, the spaces being the same, the times will be inversely in that proportion.

The deductions, therefore, are

1. That the product of the weight and height of plane, divided by the length of plane, gives the requisite power to sustain or balance the weight.

2. That the product of the power and length of plane, divided by the height of plane, (which reverses the former process,) gives the weight or resistance that the power will overcome.

3. That (the times being equal) - the velocity attained, or force acquired, or space described, by a body falling freely from rest, multiplied by the height of plane, affords a product which, divided by the length of plane, gives the velocity attained, or force acquired, or space described, by a body moving down the plane, impelled by its own gravity.

4. That the product of the weight and base of plane, divided by the length of plane, gives the pressure on the plane.

-

To find the base of the plane.

RULE. - From the square of the length of the plane, subtract the square of the height, and the square root of the difference is the base.

To find the height of the plane.

RULE. From the square of the length of the plane subtract the square of the base, and the square root of the difference is the height,

RULE.

To find the length of the plane.

- Add the square of the base and the square of the height together, and find the square root of the sum, which will be the length sought.

WEDGE.

The WEDGE is a double inclined plane. Its principles are the same, and they are wholly covered by the preceding.

The power multiplied by the length of a side, equals the resistance multiplied by half the breadth of the head. When, therefore, both sides of the substance to be cleft are movable, the product of the resistance and half the breadth of the head, divided by the length of the side of the wedge, gives the requisite force to be applied. And when only one side of the substance is movable, the product of the resistance and breadth of the head, divided by the length of a side of the wedge, gives the power required.

SCREW.

If we take the figure of an inclined plane-a right-angled triangle say, cut from paper-and unite the extremities of the base, we have the figure of the screw, in principle; and the principle of the screw is that of an inclined plane curved to a cylinder; and the screw is not a mechanical power, any more than the wedge, or wheel and axle. It is the PLANE that is an element of machinery, and not the curve or the cylinder around which the plane is placed. And it appears that the screw, the inclined plane, and the rightangled triangle, are mathematically the same. Thus, if we would find the length of the thread of a screw by the circumference and pitch, we are to find it as we would find the length of the inclined plane by the base and height, or the hypotenuse of a triangle by the base and perpendicular, and so, in like manner, for the other lines of the figure.

The pitch of the screw or rise of the thread in a revolution corresponds, to the height of the plane or perpendicular of the triangle. The circumference of the screw corresponds to the base of the plane or base of the triangle. And the length of the thread making one revolution around the cylinder the working circumference of the corresponds to the length of the plane or hypotenuse of the triangle. The mechanical advantage of a screw is as the length of the plane to the size of its angle of inclination.

screw —