The Mechanical Powers are: the lever, inclined plane, wheel and axle, the wedge, pulley, and the screw.

319c. Levers are either straight or bent, and are of three kinds.

LEVERS CONSIDERED WITHOUT WEIGHT.

Lever of the first kind is when the power, P, and weight, W, are on opposite sides of the fulcrum, F. Then P: W:: AF: B F, which is true for the three kinds of levers, and from which we find PXBF=WX AF. PX BF

P =

and W =

WXAF
?
BF
WXAF

BF=

.

(See Fig. I.)

AF

and A F=

P

PX BF
W

Lever of the second

the power, (Fig. II.)

kind is when the weight is between the fulcrum and Then P: W:: A F to B F, as above.

Lever of the third kind (Fig. III.) is when the power is between the fulcrum and the weight. Then P: W:: AF: BF, as above.

Hence, we have the general rule: The power is to the weight as the distance from the weight to the fulcrum, is to the distance from the power to the fulcrum.

In a bent lever (Fig. IV.), instead of the distances A F and F B, we have to use Fa and Fb. Then P: W:: Fa: Fb; or, P: W:: FAX cos. <AFa: FBX cos. <BF b.

Let PA BW represent a lever (see Fig. V.) Produce PA and W B to meet in C. Now the forces P and W act on C; their resultant is C R, passing through the fulcrum at F.

Let A Fa, BF b, <PA B = n, and < ABW=

319d.

LEVERS HAVING WEIGHT.

When the lever is of the same uniform size and weight. Let A B =

a lever whose weight is w.

(Fig. VI.)

Case 1. Let the centre of gravity, f, be between the fulcrum, F, and power, P; then we have, by putting F f=d, W.AFP. BF+dw,

W.AF-dw

P =

BF

Case 2. When the centre of gravity, f, is between the fulcrum and the weight.

Then WAF+dw= =P.BF.

W =

P.BF dw

AF

BF

=

=

Example from Baker's Statics. Let the length of the lever 8 feet, A F 3;.. B F 5, its weight 4 lbs., and W suspended at A = 100 lbs. Required the weight P suspended at B, the beam being uniform in all respects. We have the centre of gravity, a, = 4 feet from A, and at 1 foot from F towards P. Then, by case 1,

P =

W AF-dw 100

BF

319e. Carriage wheel meeting an obstruction (see Fig. VII.) is a lever of the first kind, where the wheel must move round C. Let DWC a wheel whose radius =abcd W. The angle of draught, P Q W, a, and C, the obstruction, whose height = h. Let Cn and Cm be drawn at right angles, to O W and O P. Then Cm represents the power, and C n the weight; then P: W:: Cn: Cm : sine COn: sine C O m.

=

DW2r; . . D n = 2 r - h; and by Euclid, B. 2, prop. 5, (2 rh) h + On2 C 02.

When the line of draught is parallel to the road, then C mrFrom this we have P: W:: V2rh-h2: r - h,

√(2 rh-h2)

And P = W.

.

A general formula.

r-h

Example. A loaded wagon, having a load of 3200 lbs., weight of wagon 800, meets a horse-railroad, whose rails are 3 inches above the street, the diameter of the wheel being 60 inches. Require the resistance or necessary force to overcome this obstacle.

Total weight of wagon and load, 4000 lbs. Weight on one wheel, 2000. 160 X 3-9 30-3

... P2000 X

968.9 lbs., which is about three times

the force of a horse drawing horizontally from a state of rest.

Hence appears the injustice of punishing a man because he cannot leave a horse-railroad track at the sound of a bell, and the necessity of the local authorities obliging the railroad companies to keep their rail level with the street or road.

Of the Inclined Plane.

=

319f. Let the base, A B, b, height, B C, h, and length, A C, = 1. The line of traction or draught must be either parallel to the base, A B, as W P' parallel to the slant, or the inclined plane, as W P, or make an angle a with the line C W, W being a point on the plane where the centre of pressure of the load acts.

When the power P' acts parallel to the base, we have

P: W:: BC: BA::h: b; or,

P: W:: sine <BAC : sine < A C B.

When the line of traction makes an angle a with the slant, then

P//: W:: sine <BA P: cos. <P// W C, from which, by alternation and inversion, we can find either quantity.

=

Example. W 20000 fbs., <BAC = 6o, < P// W C = 4°. Required the sustaining power, P.

sine B A P

P//=

P// W C

W sine B A P
Cos.<P//W C

sine 4°

.06976

= W.

= W.

Of the Wheel and Axle.

319g. When the axle passes through the centre of the wheel at right angles to its plane, and that a weight, W, is applied to the axle, and the power, P, applied to the circumference, there will be an equilibrium, when the power is to the weight as the radius of the axle is to the radius of the wheel. Let R radius of the wheel, and r = radius of the axle, both including the thickness of the rope; then we have P: W::r: R; from which we have

=

=

PR and r =

W

(A.)

Compound Axle is that which has one part of a less radius than the other. A rope and pulley is so arranged that in raising the weight, W, the rope is made to coil on the thickest part, and to uncoil from the thinAn equilibrium will take place, when 2 P⚫ D W (R − r). D= distance of power from the centre of motion. R = radius of thicker part of axis, and r that of the thinner.

ner.

319h. Toothed Wheels and Axles or Pinions. Let a, b and c be three axles or pinions, and A, B and C, three wheels.

The number of teeth in wheels are to one another as their radii. P: W:: abc: ABC: that is, the power is to the weight as the product of all the radii of the pinions is to the product of all the radii of the wheels. Or, P is to W, as the product of all the teeth in the pinions is to the product of all the teeth in the wheels.

(B.) Example 1. A weight 2000 lbs. is sustained by a rope 2 inches in diameter, going round an axle 6 inches in diameter, the diameter of the wheel being 8 feet.

Example 2. In a combination of wheels and axles there are given the radii of three pinions, 4, 6 and 8 inches, and the radii of the corresponding wheels, 20, 30 and 40 inches. What weight will P= 100 lbs. sustain at the circumference of the axle or last pinion.

319%. The power of the wedge increases as its angle is acute. In tools for splitting wood, the AC B 30°, for cutting iron, 50°, and for brass, 60°.

P: W::AB: AC; or,

P: W:: 2 sine A C B: 1.

=

Of the Pulley. (See next Fig.)

319. The pulley is either fixed or moveable.

In a fixed pulley (Fig. I.), the power is equal to the weight.

In a single moveable pulley (Fig. II.), the rope is made to pass under the lower pulley and over the upper fixed one. Then we have P: W:: 1: 2. When the upper block or sheeve remains fixed, and a single rope is made to pass over several pulleys (Fig. IV.)-for example, n pulleys-then

W

P: W:1: n, and P n = W, and P =, so that when n= 6, the power will be one-sixth of the weight.

n

When there are several pulleys, each hanging by its own cord, as in Fig. III., P: W:: 1:2 n.

Here n denotes the number of pulleys.

1600 fbs., n = 4 pulleys. Then PX24 = W;

that is, PX 16 = 1600, and P = 100 lbs.

319k.

Of the Screw.

Let LD= = distance between the threads, and r = radius of the

power from the centre of the screw.

P: W::d: 6.2832 r.

PrX 6.2832 = W D.

Then

Wd

and P =

6.2832 r

Example. Given the distance, 70 inches, from the centre of the screw to a point on an iron bar at which he exerts a power of 200, the distance between the contiguous threads 2 inches, to find the weight which he can raise. Here r = 70, d= 2, and P = 200 lbs.

VIRTUAL VELOCITY.

319m. In the Lever, P: W:: velocity of W: velocity of P.

In the Inclined Plane, vel. P: vel. W:: distance drawn on the plane : the height raised in the same time.

Let the weight W be moved from W to a, and raised from o to a; then vel. P. vel. W:: Wa: o a. (Fig. VIII.)

:

In the Wheel and Axle, vel. P: vel. W:: radius of axle: rad. of wheel : W: P.

In the single Moveable Pulley, vel. P: vel. W:: 2 : 1 :: W: P.

In a system of Pulleys, vel. P: vel. W:: n: 1 :: W: P. Here n = number of ropes.

In the ARCHIMEDEAN Screw, vel. P: vel. W, as the radius of the power multiplied by 6.2832 is to the distance between two contiguous threads. Let R radius of power, and d distance between the threads; then vel. P vel. W:: 6.2832 R: d:: W: P.

=

=

OF FRICTION.

319n. Friction is the loss due to the resistance of one body to another moving on it. There are two kinds of friction-the sliding and the rolling. The sliding friction, as in the inclined plane and roads; the rolling, as in pulleys, and wheel and axle.

Experiments on Friction have been made by Coulomb, Wood, Rennie, Vince, Morin, and others.

Those of Morin, made for the French Government, are the most extensive, and are adopted by engineers. When no oily substance is interposed between the two bodies, the friction is in proportion to their perpendicular pressures, to a certain limit of that pressure. The friction of two bodies pressed with the same weight is nearly the same without regard to the surfaces in contact. Thus, oak rubbing on oak, without unguent, gave a coefficient of friction equal to 0.44 per cent.; and when the surfaces in contact were reduced as much as possible, the coefficient was 0.41.

Coulomb has found that oak sliding on oak, without unguent, after a few minutes had a friction of 0.44, under a vertical pressure of 74 lbs.; and that by increasing the pressure from 74 to 2474 lbs., the coefficient of friction remained the same.

Friction is independent of the velocities of the bodies in motion, but is dependent on the unguents used, and the quantity supplied.

Morin has found that hog's lard or olive oil kept continuously on wood moving on wood, metal on metal, or wood on metal, have a coefficient of 0.07 to 0.08; and that tallow gave the same result, except in the case of metals on metals, in which case he found the coefficient 0.10.

Different woods and metals sliding on one another have less friction. Thus, iron on copper has less friction than iron on iron, oak on beach has less than oak on oak, etc.

=

The angle of friction is <BAC, in the annexed figure, where W represents the weight, kept on the inclined plane AC by its friction. Let G = centre of gravity; then the line I K represents the weight W, in direction of the line of gravity, which is perpendicular to A B; IL the pressure perpendicular to A C,

and IN L K = the friction or weight sufficient to keep the weight W on the plane. The two triangles, A B C and I K L are similar to one