for rapid motions, and light pressures, oils are generally employed.

The ratio obtained by dividing the entire force of friction by the normal pressure, is called the coefficient of friction; the value of the coefficient of friction for any two substances, may be determined experimentally as follows:

ΑΓ

Р

Fig. 92.

Let AB be a horizontal plane formed of one of the substances, and let O be a cubical block of the other substance resting upon it. Attach a string OC, to the block, so that its direction shall pass through its centre of gravity, and be parallel

to AB; let the string pass over a fixed pulley C, and let a weight F, be attached to its extremity.

Increase the weight F till the body O just begins to slide along the plane, then will this weight measure the whole force of friction. Denote this weight by F, that of the body, or the normal pressure, by P, and the coefficient of friction, by ƒ. Then, from the definition, we shall have,

In this manner, values for f, corresponding to different substances, may be found, and arranged in tables. This experiment gives the friction of quiescence. If the weight Fis such as to keep the body O in uniform motion, the resulting value of ƒ will correspond to friction of motion.

The value of f, for any substance, is called the unit, or coefficient of friction. Hence, we may define the unit, or coefficient of friction, to be the friction due to a normal pressure of one pound.

Having given the normal pressure in pounds, and the unit of friction, the entire friction will be found by multiplying these quantities together.

There is a second method of finding the value of ƒ experimentally, as follows:

a

Р

B

Let AB be an inclined plane, formed of one of the substances, and O a cubical block, formed of the other substance, and resting upon it. Elevate the plane till the block just begins to slide down the plane by its own weight. Denote the angle of inclination, at this instant, by a, and the weight of O, by W. Resolve the force W into two components, one normal to the surface of the plane, and the other one parallel to it. Denote the former component by P, and the latter by Q. Since OW is perpendicular to AC, and OP to AB, the angle WOP is equal to a. Hence,

FWcosa, and Q Wsina.

=

Fig. 93.

The normal pressure being equal to Wcosa, and the force of friction being Wsina, we shall have, from the principles already explained,

The angle a is called the angle of friction.

Limiting Angle of Resistance.

103. Let AB be any plane surface, and O a body resting upon it. Let R be the resultant

of all the forces acting upon it, including the weight applied at the centre of gravity. Denote the angle between R and the normal to AB, ·

by a, and suppose R to be resolved into two components P and Q, the

P

Ꭺ

Β ́

Fig. 94.

former parallel to AB, and the latter perpendicular to it;

we shall have,

P Rsina, and Q
=

= Rcosa.

The friction due to the normal pressure will be equal to fRcosa. Now, when the tangential component Psing is less than fRcosz, the body will remain at rest; when it is greater than fRcosz, the body will slide along the plane; and when the two are equal, the body will be in a state bordering on motion along the plane. Placing the two equal, we have,

The value of a is called the limiting angle of resistance,

and is equal to

the inclination of the

0

plane, when the body is about to slide down by its own weight. If, now, the line OR be revolved about the normal, it will generate a conical surface, within which, if any force whatever, including the weight, be applied at the centre of gravity, the body will remain at rest, and without which, if a sufficient force be applied, the body will slide along the plane. This cone is called the limiting cone of resistance.

Fig. 95.

The values of f, or the coefficient of friction, in some of the most common cases, as determined by MORIN, is appended:

Bodies between which friction takes place.
Leather belts on wooden pulleys,
Leather belts on cast iron pulleys,

Coefficient of friction.

.47

.28

.10

Cast iron on cast iron, greased,

Pivots or axes of wrought or cast iron, on brass or cast

iron pillows:

104. Rolling friction is the resistance which one body offers to another when rolling along its surface, the two being pressed together by some force. This resistance, like that in sliding friction, arises from the inequalities of the two surfaces. The coefficient, or unit, of rolling friction is equal to the quotient obtained by dividing the entire force of friction by the normal pressure. This coefficient is much less than the coefficient of sliding friction.

The following laws of friction have been established, when a cylindrical body or wheel rolls upon a plane:

First, the coefficient of rolling friction is proportional to the normal pressure:

Secondly, it is inversely proportional to the diameter of the cylinder or wheel:

Thirdly, it increases as the surface of contact and velocity increase.

In many cases there is a combination of both sliding and rolling friction in the same machine. Thus, in a car upon a railroad-track, the friction at the axle is sliding, and that between the circumference of the wheel and the track is rolling.

Adhesion.

105. ADHESION is the resistance which one body experiences in moving upon another in consequence of the cohesion existing between the molecules of the surfaces in contact. This resistance increases when the surfaces are

allowed to remain for some time in contact, and is very slight when motion has been established. Both theory and experiment show that adhesion between the same surfaces is proportional to the extent of the surface of contact.

The coefficient of adhesion is the quotient obtained by dividing the entire adhesion by the area of the surface of contact. Or, denoting the entire adhesion by A, the area of the surface of contact by S, and the coefficient of adhesion by a, we have,

To find the entire adhesion, we multiply the unit of adhesion by the area of the surface of contact.

Stiffness of Cords.

106. Let O represent a pulley, with a cord AB, wrapped around its circumference, and suppose a force P, applied at B, to overcome the resistance R, and impart motion to the pulley. As the rope winds upon the pulley, at C, its rigidity acts to increase the arm of lever of R, and to overcome this resistance to flexure an additional force is required. For the same pulley, this additional force may be represented by the algebraic expression,

a+bR,

R

Fig. 96.

A

in which a and b are constants dependent upon the nature and construction of the rope, and R is the resistance to be overcome, or the tension of the cord A C. The values of a and b for different ropes have been ascertained by experiment, and tabulated. Finally, if the same rope be wound upon pulleys of different diameters, the additional force is found to vary inversely as their diameters. If the diameter of the pulley be denoted by D, and the resistance due to stiffness of cordage be denoted by S, we shall have,