In the case of the pulley, if we neglect friction, we shall have, when the motion is uniform,

for the algebraic expression of the conditions of equilibrium. The values of a and b have been determined experimentally for all values of R and D, and tabulated.

Atmospheric Resistance.

107. The atmosphere exercises a powerful resistance to the motion of bodies passing through it. This resistance is due to the inertia of the particles of air, which must be overcome by the force of a moving body. It is evident, in the first place, other things being equal, that the resistance will depend upon the amount of surface of the moving body which is exposed to the air in the direction of the motion. In the second place, the resistance must increase with the square of the velocity of the moving body; for, if we suppose the velocity to be doubled, there will be twice as many particles met with in a second, and each particle will collide against the moving body with twice the force, hence; if the velocity be doubled, the resistance will be quadrupled. By a similar course of reasoning, it may be shown that, if the velocity be tripled, the retardation will become nine times as great, and so on. If, therefore, the retardation corresponding to a square foot of surface, at any given velocity, be determined, the retardation corresponding to any surface and any velocity whatever may be computed.

Influence of Friction on the Inclined Plane.

108. Let it be required to determine the relation between the power and resistance, when the power is just on the point of imparting motion to a body up an inclined plane, friction being taken into account.

Let AB represent the plane, O the body, OP the power on the point of imparting motion

α

P

R

R

R

up the plane, and OR the weight of the body. Denote the power by P, the weight by R, the inclination of the plane by a, and the angle between the direction of the power and the normal to the plane by B. Let P and R be resolved into components respectively parallel and perpendidicular to the plane. We shall have, for the parallel components, Rsina and Psinß, and for the perpendicular components, Rcosa and Pcos. The resultant of the normal components will be equal to Rcosa - Pcosß; and, if we denote the coefficient of friction by ƒ, we shall have for the entire force of friction (Art. 102),

f(Reosx- Pcos3).

Fig. 97.

When we consider the body on the eve of motion up the plane, the component Psing must be equal and directly opposed to the resultant of the force of friction and the component Rsina; hence, we must have,

Psinß = Rsina +ƒ (Rcosë

Pcosß).

Performing the multiplications indicated, and reducing, we have,

P = R{ sina + fcosa
sing + fes

If we suppose an equilibrium to exist, the body being on on the eve of motion down the plane, we shall have.

From these expressions, two values of P may be found, when a, B, f, and R are given. It is evident that any value of P greater than the first will cause the body to slide up the plane, that any value less than the second will permit it to slide down the plane, and that for any intermediate value the body will remain at rest on the plane. If we suppose P to be parallel to the plane, we shall have 0, and the two values of P reduce to

sinß

=

and,

1, cosß

=

If friction be neglected, we have f= 0; whence, by substitution,

P BC

Р P= Rsina, or

a result which agrees with that deduced in a preceding article.

To find the quantity of work of the power whilst drawing a body up the entire length of the inclined plane, it may be observed that the value of P, in Equation (53), is equal to that required to maintain the body in uniform motion after motion has commenced.

Multiplying both members of that equation by AB, we

have,

P× AB = R x AB sin a + fR × AB cosa

= R × BC + ƒR × AC.

But RBC is the quantity of work necessary to raise the body through the vertical height BC; and fR × AC, is the quantity of work necessary to draw the body horizontally through the distance AC (Art. 75). Hence, the quantity of work required to draw a body up an inclined plane, when the power is parallel to the plane, is equal to the quantity of work necessary to draw it horizontally across the base of the plane, plus the quantity of work necessary to raise it vertically through the height of the plane.

A curve situated in a vertical plane may be regarded as made up of an infinite number of inclined planes. We infer, therefore, that the quantity of work necessary to draw a body up a curve, the power acting always parallel to the direction of the curve, is equal to the quantity of work necessary to draw the body over the horizontal projection of the curve, plus the quantity of work necessary to raise the body through a height equal to the difference of altitude of the two extremities of the curve.

The last two principles enable us to compare the quantities of work necessary to draw a train of cars over a horizontal track, and up an inclined track, or a succession of inclined tracks. We may, therefore, compute the length of a horizontal track which will consume the same amount of work, furnished by the motor, as is actually consumed in consequence of the undulation of the track.

We are thus enabled to compare the relative advantages of different proposed routes of railroad, with respect to the motive power required for working them.

Line of Least Traction.

109. The force employed to draw a body with uniform motion along an inclined plane, is called the force of traction; and the line of direction of this force is the line of traction. In Equation (51), P represents the force of traction required to keep a body in uniform motion up an inclined plane, and 6 is the angle which the line of traction. makes with the plane. It is plain, that when 3 varies, other things being the same, the value of P will vary; there will evidently be some value of B, which will render P the least possible; the direction of P in this case, is called the line of least traction; and it is along this line that a force can be applied with greatest advantage, to draw a body up an inclined plane. If we examine the expression for P, in Equation (51), we see that the numerator remains constant; therefore, the expression for P will be least possible when the denominator is the greatest possible. By a simple pro

cess of the Differential Calculus, it may be shown that the denominator will be the greatest possible, or a maximum, when,

fcot ß, or f =

tan(90° — B).

=

That is, the power will be applied most advantageously, when it makes an angle with the inclined plane equal to the angle of friction.

From the second value of P, it may be shown, in like manner, that a force will be most advantageously applied, to prevent a body from sliding down the plane, when its direction makes an angle with the plane equal to the supplement of the angle of friction, the angle being estimated as before from that part of the plane lying above the body.

Friction on an Axle.

110. Let it be required to determine the position of equilibrium of a horizontal axle, resting in

a cylindrical box, when the power is just on the point of overcoming the friction between the axle and box.

00

Let O' be the centre of a cross section of the axle, O the centre of the cross section of the box, and N their point of contact, when the power is on the point of overcoming the friction between the axle and box. The element through N will be the line of contact of the axle and box.

Fig. 98.

When the axle is only acted upon by its own weight, the element of contact will be the lowest element of the box. If, now, a power be applied to turn the axle in the direction indicated by the arrow-head, the axle will roll up the inside of the box until the resultant of all the forces acting upon it becomes normal to the surface of the axle at some point of the element through N. This normal force pressing the axle against the box, will give rise to a force of friction acting tangentially upon the axle, which will be exactly equal to the tangential force applied at the circumference of the