Mechanics for engineers - Morley

Engineering students constitute a fairly large proportion of those attending the Mechanics classes in technical colleges and schools, but their needs are not identical with those of the students of general science. It has recently become a common practice to provide separate classes in Mathematics, adapted to the special needs of engineering students, who are in most institutions sufficiently numerous to justify similar provision in Mechanics. The aim of this book is to provide a suitable course in the principles of Mechanics for engineering students.

With this object in view, the gravitational system of units has been adopted in the English measures. A serious injustice is often done to this system in books on Mechanics by wrongly defining the pound unit of force as a variable quantity, thereby reducing the system to an irrational one. With proper premises the gravitational system is just as rational as that in which the "poundal" is adopted as the unit of force, whilst it may be pointed out that the use of the latter system is practically confined to certain text-books and examination papers, and does not enter into any engineering work. Teachers of Engineering often find that students who are learning Mechanics by use of the "poundal" system, fail to apply the principles to engineering problems stated in the only units which are used in such cases the gravitational units. The use of the dual system is certainly confusing to the student, and in addition necessitates much time being spent on the re-explanation of principles, which might otherwise be devoted to more technical work.

Graphical methods of solving problems have in some cases been used, by drawing vectors to scale, and by estimating slopes and areas under curves. It is believed that such exercises, although often taking more time to work than the easy arithmetic ones which are specially framed to give exact numerical answers, compel the student to think of the relations between the quantities involved, instead of merely performing operations by fixed rules, and that the principles so illustrated are more deeply impressed.

The aim has not been to treat a wide range of academic problems, but rather to select a course through which the student may work in a reasonable time say a year and the principles have been illustrated, so far as the exclusion of technical knowledge and terms would allow, by examples likely to be most useful to the engineer.

In view of the applications of Mechanics to Engineering, more prominence than usual has been given to such parts of the subject as energy, work of forces and torques, power, and graphical statics, while some other parts have received less attention or have been omitted.

It is usual, in books on Mechanics, to devote a chapter to the equilibrium of simple machines, the frictional forces in them being considered negligible: this assumption is so far from the truth in actual machines as to create a false impression, and as the subject is very simple when treated experimentally, it is left for consideration in lectures on Applied Mechanics and in mechanical laboratories.

The calculus has not been used in this book, but the student is not advised to try to avoid it; if he learns the elements of Mechanics before the calculus, dynamical illustrations of differentiation and integration are most helpful. It is assumed that the reader is acquainted with algebra to the progressions, the elements of trigonometry and curve plotting; in many cases he will doubtless, also, though not necessarily, have some little previous knowledge of Mechanics.

The ground covered is that required for the Intermediate (Engineering) Examination of the University of London in Mechanics, and this includes a portion of the work necessary for the Mechanics Examination for the Associateship of the Institution of Civil Engineers and for the Board of Education Examination in Applied Mechanics.

CHAPTER III - WORK, POWER, AND ENERGY

51. Work. When a force acts upon a body and causes motion, it is said to do work.

In the case of constant forces, work is measured by the product of the force and the displacement, one being estimated by its component in the direction of the other.

One of the commonest examples of a force doing work is that of a body being lifted against the force of gravity, its weight. The work is then measured by the product of the weight of the body, and the vertical height through which it is lifted. If we draw a diagram (Fig. 33) setting off the constant force F by a vertical ordinate, OM, then the work done during any displacement represented by ON is proportional to the area MPNO, and is represented by that area.

52. Units of Work. Work being measured by the product of force and length, the unit of work is taken as that done by a unit force acting through unit distance. In the British gravitational or engineer's system of units, this is the work done by a force of 1 lb. acting through a distance of 1 foot. It is called the foot-pound of work. If a weight W lbs. be raised vertically through h feet, the work done is Wh foot-lbs.

Occasionally inch-pound units of work are employed, particularly when the displacements are small.

In the C.G.S. system the unit of work is the erg. This is the work done by a force of one dyne during a displacement of 1 centimetre in its own direction (see Art. 42).

53. Work of a Variable Force. If the force during any displacement varies, we may find the total work done approximately by splitting the displacement into a number of parts and finding the work done during each part, as if the force during the partial displacement were constant and equal to some value it has during that part, and taking the sum of all the work so calculated in the partial displacements. We can make the approximation as near as we please by taking a sufficiently large number of parts. We may define the work actually done by the variable force as the limit to which such a sum tends when the subdivisions of the displacement are made indefinitely small.

54. Graphical Representation of Work of a Variable Force. Fig. 34 is a diagram showing by its vertical ordinates the force acting on a body, and by its horizontal ones the dis- placements. Thus, when the displacement is represented by ON, the force acting on the body is represented by PN. Suppose the interval ON divided up into a number of small parts, such as CD. The force acting on the body is represented by AC when the displacement is that represented by OC. Since the force is increasing with increase of displacement the work done during the displacement CD is greater than that represented by the rectangle AEDC, and less than that represented by the rectangle FBDC. The total work done during the displacement will lie between that represented by the series of smaller rectangles, such as AEDC, and that represented by the series of larger rectangles, such as FBDC. The area MPNO under the curve MP will always lie between these total areas, and if we consider the number of subdivisions of ON to be carried higher indefinitely, the same remains true both of the total work done and the area under the curve MP. Hence the area MPNO under the curve MP represents the work done by the force during the displacement represented by ON.

The Indicator Diagram, first introduced by Watt for use on the steam-engine, is a diagram of the same kind as Fig. 34. The vertical ordinates are proportional to the total force exerted by the steam on the piston, and the horizontal ones are proportional to the displacement of the piston. The area of the figure is then proportional to the work done by the steam on the piston.

Average Force. The whole area MPNO (Figs. 34 and 35) divided by the above ON gives the mean height of the area; this represents the space-average of the force during the displacement ON. This will not necessarily be the same as the time-average (Art. 45). We may define the space-average of a varying force as the work done divided by the displacement.

55. Power. Power is the rate of doing work, or the work done per unit of time.

One foot-pound per second might be chosen as the unit of power. In practice a unit 550 times larger is used; it is called the horse-power. It is equal to a rate of 550 foot-lbs. per second, or 33,000 foot-lbs. per minute. In the C.G.S. system the unit of power is not usually taken as one erg per second, but a multiple of this small unit. This larger unit is called a watt, and it is equal to a rate of 107 ergs per second. Engineers frequently use a larger unit, the kilowatt, which is 1000 watts. One horse-power is equal to 746 watts or 0,746 kilowatt.