at any instant. As an example, suppose the first wheel to have 100 teeth, and to bear on its arbor a smaller wheel, having 10 teeth; suppose this wheel to engage with a larger wheel having 100 teeth, and so on. When the endless screw has made 1,000 revolutions, the first wheel will have made 100 revolutions, the second large wheel will have made 10 revolutions, and the third wheel 1 revolution. By a suitable arrangement of indices, the exact number of revolutions of the axis, at any instant, may be read off from the instrument.

EXAMPLES.

1. What must be the distance between the threads of a screw in order that a power of 28 lbs., acting at the extremity of a lever 25 inches long, may sustain a weight of 10,000 lbs. ? Ans. .4396 inches.

2. The distance between the threads of a screw is of an inch. What resistance can be supported by a power of 60 lbs., acting at the extremity of a lever 15 inches long? Ans. 16,964 lbs.

3. The distance from the axis of the trunions of a gun weighing 2,016 lbs. to the elevating screw is 3 feet, and the distance of the centre of gravity of the gun from the same axis is four inches. If the distance between the threads of the screw be of an inch, and the length of the lever 5 inches, what power must be applied to sustain the gun in a horizontal position? Ans. 4.754 lbs.

100.

The Wedge.

The wedge is a solid, bounded by a rectangle

BD, called the back; two equal rect-
angles, AF and DF, called faces;
and two equal isosceles triangles, called
ends.
The line EF, in which the
faces meet, is called the edge.

The power is applied at the back, to which its direction should be normal, and the resistance is applied to the faces, and in directions normal to them. One half of the resistance

D

C

E

Fig. 89.

Br

Р

R

R

D

A

Fig. 90.

is applied normally to one face, and the other half normally to the other face. Let ABC be a section of a wedge made by a plane at right angles to the edge. Denote the power by P, and the resistance opposed to each face by R; denote the angle BAC of the wedge by 2p. Produce the directions of the resistances till they intersect in 0. This point will be on the line of direction of the power. Lay off OF to represent the power, and complete the parallelogram ED; then will OE and sent the resistances developed by the power. of the forces R be resolved into two components, one perpendicular to OF, and the other coinciding with it. The two former will be equal and directly opposed to each other, whilst the two latter will hold the force P in equilibrium. Since DE is perpendicular to FO, and DO perpendicular to CA, the angle ODE is equal to the angle QAC, or . The component of R in the direction of OF, is Rsinp; hence, twice this, or Rsing = P. But CK

sin p

=

CA

OD repre-
Let each

= in which b denotes the breadth of the

back BC, and the length of the face CA. Substituting this expression for sing, and reducing, we have,

:

Rxb Pl, or PR:16 7. (49.)

That is, the power is to the resistance as one-half of the breadth of the back is to the length of the face of the wedge.

The mechanical advantage of the wedge may be increased by diminishing the breadth of the back, or, in other words, by making the edge sharper. The principle of the wedge finds an important application in all cutting instruments, as knives, razors, and the like. By diminishing the thickness of the back, the instrument is rendered liable to break, hence the necessity of forming cutting instruments of the hardest and most tenacious materials.

101.

General remarks on Elementary Machines.

We have thus far supposed the power and resistance to be in equilibrium, through the intervention of the machine, their points of application being at rest. If we now suppose the point of application to be moved through any distance, by the action of an extraneous force, the point of application of the power will move through a corresponding space. These spaces will be described in conformity with the design of the machine; and it will be found, in each instance, that they are inversely proportional to the forces. If we suppose these spaces to be infinitely small, they may, in all cases, be regarded as straight lines, which will also be the virtual velocities of the forces. If the point of application moves in a direction contrary to the direction of the resistance, the point of application of the power will move in the direction of the power. If we denote the paths described by those points respectively, by dr, and dp, we shall have,

R&r=0; or Pop

= Rdr (50.)

That is, the algebraic sum of the virtual moments is equal to 0. Or, we might enunciate the principle in another manner, by saying, that in all cases, the quantity of work of the power is equal to the quantity of work of the resistance.

We shall illustrate this principle, by considering a single case, that of the single movable pulley, leaving its further application to the remaining machines, as exercises for the student.

In the figure, suppose that an extraneous force acts to raise the resistance R, through the infinitely small space DE, denoted by dr; the point of application of P must be raised through the infinitely small space FG, denoted by op, in order that the equilibrium may be preserved.

In order that the resistance may be raised through the distance DE, both branches of the rope enveloping the pulley must be shortened by the same amount; or, what is the same

6*

E

ען

R
Fig. 91.

thing, the free end of the rope must ascend through twice the distance DE. Hence,

ὁρ

= 2ôr.

But, from the conditions of equilibrium,

P = }R.

Multiplying these equations, member by member, we have,

Pop = Rôr.

In

Hence, the principle is proved for this particular case. like manner, it may be shown to hold good for all of the elementary machines.

The principle of equality of work of the power and resistance being true for any infinitely short time, it must necessarily hold good for any time whatever. Hence, we conclude, that the quantity of work of the power, in overcoming any resistance, is equal to quantity of work of the resistance. Although, by the application of a very small power, we are able to overcome a very great resistance, the space passed over by the point of application of the power must be as much greater than that passed over by the point of application of the resistance, as the resistance is greater than the power. This is generally expressed by saying, that what is gained in power is lost in velocity.

We see, therefore, that no power is, or can be, gained; the only function of a machine being to enable a smaller force to accomplish in a longer time, what a larger force would be required to perform in a shorter time.

Friction.

102. FRICTION is the resistance which one body experiences in moving upon another, the two being pressed together by some force. This resistance arises from inequalities in the two surfaces, the projections of one surface sinking into the depressions of the other. In order to overcome this resistance, a sufficient force must be applied

to break off, or bend down, the projecting points, or else to lift the moving body clear of the inequalities. The force thus applied, is equal, and directly opposed to the force of friction, which is tangential to the two surfaces. The force which presses the surfaces together, is normal to them both at the point of contact.

Friction is distinguished as sliding and rolling. The former arises when one body is drawn upon another; the latter when one body is rolled upon another. In the case of rolling friction, the motion is such as to lift the projecting points out of the depressions; the resistance is, therefore, much less than in sliding friction.

Between certain bodies, the friction is somewhat different when motion is just beginning, from what it is when motion has been established. The friction developed when a body is passing from a state of rest to a state of motion, is called friction of quiescence; that which exists between bodies in motion, is called friction of motion.

The following laws of friction have been established by numerous experiments, viz. :

First, the friction of quiescence between the same bodies, is proportional to the normal pressure, and independent of the extent of the surfaces in contact.

Secondly, the friction of motion between the same bodies, is proportional to the normal pressure, and independent, both of the extent of the surfaces in contact, and of the velocity of the moving body.

Thirdly, for compressible bodies, the friction of quiescence is greater than the friction of motion; for bodies which are sensibly incompressible, the difference is scarcely appreciable.

Fourthly, friction may be greatly diminished, by interposing unguents between the rubbing surfaces.

Unguents serve to fill up the cavities of surfaces, and thus to diminish the resistances arising from their roughness. For slow motions and great pressures, the more consistent unguents are used, as lard, tallow, and various mixtures;