velocity of 2 feet. How far must the body be situated from the axis that it may be on the point of sliding outwards, the coefficient of friction between the body and plane being equal to .6 ?

SOLUTION.

Denote the required distance by r; then will the velocity of the body be equal to 2r, and the acceleration due to the centrifugal force will be equal to 4r. But the acceleration due to the force of friction is equal to 0.6 × g 19.3 ft. From the conditions of the problem, these two are equal, hence,

4r 19.3 ft.,

=

r = 4.825 ft. Ans.

6. What must be the elevation of the outer rail of a railroad track, the radius of curvature being 3960 ft., the distance between the rails 5 feet, and the velocity of the car 30 miles per hour, in order that the centrifugal force may be exactly counterbalanced by the component of the weight. parallel to the line joining the rails?

Ans. 0.076 ft., or 0.9 in., nearly.

7. The distance between the rails is 5 feet, the radius of the curve 600 feet, and the height of the centre of gravity of the car 5 feet. What velocity must be given to the car that it may be on the point of being overturned by the centrifugal force, the rails being on the same level?

143. By the term work, in mechanics, is meant the effect produced by a force in overcoming a resistance, such as weight, inertia, &c. The idea of work implies that a force is continually exerted, and that the point at which it is applied moves through a certain space. Thus, when a weight is raised through a vertical height, the power which

overcomes the resistance offered by the weight is said to work, and the amount of work performed evidently depends, first, upon the weight raised, and, secondly, upon the height through which it is raised. All kinds of work may be assimilated to the raising of a weight. Hence it is, that this kind of work is assumed as a standard to which all other kinds of work are referred.

The unit of work most generally adopted in this country, is the effort required to raise one pound through a height of one foot. The number of units of work required to raise any weight to any height will, therefore, be equal to the product obtained by multiplying the number of pounds in the weight by the number of feet in the height. If we take the weight of the body as it would be at the equator, for the sake of uniformity in notation, we may regard the weight and the mass as identical (Art. 11). If we denote the quantity of work expended in raising a body, by Q, the mass of the body, by m, and the height, by h, we shall have,

Q = mh.

When very large quantities of work are to be estimated, as in the case of steam-engines and other powerful machines, a different unit is sometimes employed, called a horse power. When this unit is employed, time enters as an element. A horse power is a power which is capable of raising 33,000 lbs. through a height of one foot in one minute; that is, it is a power capable of performing 33,000 units of work in a minute of time, or 550 units of work in one second. When an engine, then, is spoken of as being of 100 horse power, it is to be understood that it is capable of performing 55,000 units of work in a second.

In general, if a force acts to overcome a resistance of m pounds, through a distance of n feet, whatever may be the cause of the resistance, or whatever may be the direction of the motion, the quantity of work will be measured by a unit of work taken mn times.

108100

If the pressure exerted by the force is variable, we may conceive the path described by the point of application to be divided into equal parts, so small that, for each part, the pressure may be regarded as constant. If we denote the length of one of these equal parts, by p, and the force exerted whilst describing this path, by P, we shall have for the corresponding quantity of work, Pp, and for the entire quantity of work denoted by Q, we shall have the sum of these elementary quantities of work; or, since p is the same for each,

The quotient obtained by dividing the entire quantity of work by the entire path, is called the mean pressure, or the mean resistance, and is evidently the force which, acting uniformly through the same path, would accomplish the same work.

P

A

D

B

Fig. 126.

Work, when the power acts obliquely to the path. 144. Let PD represent the force, and AB the path which the body D is constrained to follow. Denote the angle PDs by a, and suppose Pto be resolved into two components, one perpendicular, and the other parallel to AB. We shall have, for the former, Psina, and, for the latter, Pcosx. The former can produce no work, since, from the nature of the case, the point cannot move in the direction of the normal; hence, the latter is the only component which works. Let 8D be the space through which the body is moved in any time whatever. If we denote the pressure exerted in the direction of PD, by P, and the quantity of work, by Q, we shall have,

Let fall the perpendicular ss' from s, on the direction of the

force P. From the right-angled triangle Dss', we shall

have,

8D x cosα = s'D.

Substituting this in the preceding equation, we get,

Q = P x 8'D.

That is, the quantity of work of a force acting obliquely to the path along which the point of application is constrained to move, is equal to the intensity of the force multiplied by the projection of the path upon the direction of the force. We have supposed the intensity of the force P, to be expressed in pounds, or units of mass.

If we take the distance sD, infinitely small, s'D will be the virtual velocity of D, and the expression for the quantity of work of P will be its virtual moment (Art. 38). Hence we say that the elementary quantity of work of a force is equal to its virtual moment, and, from the principle of virtual moments, we conclude that the algebraic sum of the elementary quantities of work of any number of forces applied at the same point, is equal to the elementary quantity of work of their resultant. What is true for the elementary quantities of work at any instant, must be equally true at any other instant. Hence, the algebraic sum of all the elementary quantities of work of the components in any time whatever, is equal to the algebraic sum of the elementary quantities of work of their resultant for the same time; that is, the work of the components for any time, is equal to the work of their resultant for the same time. This principle would hardly seem to require demonstration, for, from the very definition of a resultant, it would seem to be true of necessity. If the forces are in equilibrium, the entire quantity of work will be equal to 0.

This principle finds an important application, in computing the quantity of work required to raise the material for a wall or building; for raising the material from a shaft; for raising water from one reservoir to another; and a great

variety of similar operations. In this connection, the principle may be enunciated as follows: The algebraic sum of the quantities of work required to raise the parts of a system through any vertical spaces, is equal to the quantity of work required to move the whole system over a vertical space equal to that described by the centre of gravity of the system.

It also follows, from the same principle, that, if all the pieces of a machine which moves without friction be in equilibrium in all positions, under the action of weights suspended from different parts of the machine, the centre of gravity of the system will neither ascend nor descend whilst the machine is in motion.

Work, when a body is constrained to move upon a curve.

145. Let AB represent the curve, and suppose that the force is so taken that its line of direction shall

a

a

P

Fig. 127.

always pass through a point P. Divide the curve into elements so small that each may be taken as a straight line, and, with P as a centre, and the distances from P to the points of division as radii, describe arcs of circles. Then, denoting the force supposed constant, by P, we shall have (from Art. 144) the elementary quantity of work performed whilst the point is moving over aa', equal to P x ac, or P × bb'. In like manner, the quantity of work performed whilst the point is describing a'a" will be equal to P x b'b', and so on. Hence, by summation, we shall find the entire quantity of work performed in moving the body from B to A will be equal to P × BB'. If now we suppose the curve AB to lie in a vertical plane, and the force to be the force of gravity, the point P may be regarded as infinitely distant, the lines Pa, Pa' &c., will become vertical, and the lines a'b', a"b", will be horizontal. We may, therefore, enunciate the following principle: The quantity of work of the weight of a body