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By J. P. Den Hartog
Dover Publications, Inc.Copyright © 1948 J. P. Den Hartog
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DISCRETE COPLANAR FORCES
1. Introduction. Mechanics is usually subdivided into three parts: statics, kinematics, and dynamics. Statics deals with the distribution of forces in bodies at rest; kinematics describes the motions of bodies and mechanisms without inquiring into the forces or other causes of those motions; finally, dynamics studies the motions as they are caused by the forces acting.
A problem in statics, for example, is the question of the compressive force in the boom of the crane of Fig. 17 (page 24) caused by a load of a given magnitude. Other, more complicated problems deal with the forces in the various bars or members of a truss (Fig. 61, page 54) or in the various cables of a suspension bridge (Fig. 67, page 63). Statics, therefore, is of primary importance to the civil engineer and architect, but it also finds many applications in mechanical engineering, for example, in the determination of the tensions in the ropes of pulleys (Fig. 15, page 21), the force relations in screw jacks and levers of various kinds, and in many other pieces of simple apparatus that enter into the construction of a complicated machine.
Statics is the oldest of the engineering sciences. Its first theories are due to Archimedes (250 b.c.), who found the laws of equilibrium of levers and the law of buoyancy. The science of statics as it is known today, however, started about a.d. 1600 with the formulation of the parallelogram of forces by Simon Stevin.
Kinematics deals with motion without reference to its cause and is, therefore, practically a branch of geometry. It is of importance to the mechanical engineer in answering questions such as the relation between the piston speed and the crankshaft speed in an engine, or, in general, the relation between the speed of any two elements in complicated "kinematical" machines used for the high-speed automatic manufacture of razor blades, shoes, or zippers. Another example appears in the design of a quick-return mechanism (Fig. 146, page 168) such as is used in a shaper, where the cutting tool does useful work in one direction only and where it is of practical importance to waste as little time as possible in the return stroke. The design of gears and cams is almost entirely a problem in kinematics.
Historically, one of the first applications of the science was "James Watt's parallelogram," whereby the rotating motion of the flywheel of the first steam engine was linked to the rectilinear motion of the piston by means of a mechanism of bars (Fig. 177, page 203). Watt found it necessary to do this because the machine tools of his day were so crude that he could not adopt the now familiar crosshead-guide construction, which is "kinematically" much simpler.
From this it is seen that kinematics is primarily a subject for the mechanical engineer. The civil engineer encounters it as well but to a lesser extent, for instance in connection with the design of drawbridges, sometimes also called "bascule" bridges, where the bridge deck is turned up about a hinge and is held close to static equilibrium in all positions by a large counterweight (Problem 44, page 353). It is sometimes of practical importance to make the motion of this counterweight much smaller than the motion of the tip of the bridge deck, and the design of a mechanism to accomplish this is a typical problem in kinematics.
Finally, dynamics considers the motions (or rather, their accelerations) as they are influenced by forces. The subject started with Galileo and Newton, three centuries ago, with applications principally to astronomy. Engineers hardly used the new science before 1880 because the machines in use up to that time ran so slowly that their forces could be calculated with sufficient accuracy by the principles of statics. Two practical dynamical devices used before 1880 are Watt's flyball engine governor and the escape mechanism of clocks. These could be and were put to satisfactory operation without much benefit of dynamical theory. Shortly after 1880 the steam turbine, the internal-combustion engine, and the electric motor caused such increases in speed that more and more questions appeared for which only dynamics could provide an answer, until at the present time the large majority of technical problems confronting the mechanical or aeronautical engineer are in this category. Even the civil engineer, building stationary structures, cannot altogether remain aloof from dynamics, as was demonstrated one sad day in 1940 when the great suspension bridge near Tacoma, Washington, got into a violent flutter, broke to pieces, and fell into the water—a purely dynamical failure. The design of earthquake-resistant buildings requires a knowledge of dynamics by the civil engineer. But these are exceptions to the general rule, and it is mainly the mechanical engineer who has to deal with dynamical questions, such as the stability of governing systems, the smooth, non-vibrating operation of turbines, the balancing of internal-combustion engines, the application of gyroscopes to a wide variety of instruments, and a host of other problems.
2. Forces. Statics is the science of equilibrium of bodies subjected to the action of forces. It is appropriate, therefore, to be clear about what we mean by the words "equilibrium" and "force." A body is said to be in equilibrium when it does not move.
"Force" is defined as that which (a) pushes or pulls by direct mechanical contact, or (b) is the "force of gravity," otherwise called "weight," and other similar "field" forces, such as are caused by electric or magnetic attraction.
We note that this definition excludes "inertia" force, "centrifugal" force, "centripetal" force, or other "forces" with special names that appear in the printed literature.
The most obvious example of a pull or a push on a body or machine is when a stretched rope or a compressed strut is seen to be attached to the body. When a book rests on the table or an engine sits on its foundation, there is a push force, pushing up from the table on the book and down from the book on the table. Less obvious cases of mechanical contact forces occur when a fluid or gas is in the picture. There is a push-force between the hull of a ship and the surrounding water, and similarly there is such a force between an airplane wing in flight and the surrounding air.
If there is any doubt as to whether there is a mechanical contact force between two bodies, we may imagine them to be separated by a small distance and a small mechanical spring to be inserted between them, with the ends of the spring attached to the bodies. If this spring were to be elongated or shortened in our imaginary experiment, there would be a direct contact force. All forces that we will deal with in mechanics are direct contact forces with the exception of gravity (and of electric and magnetic forces). The mental experiment of the inserted spring fails with gravity. Imagine a body at rest suspended from above by a string. According to our definition there are two forces acting on the body: an upward one from the stretched string, and a downward one from gravity. We can mentally cut the string and insert a spring between the two pieces. This spring will be stretched by the pull in the string. We cannot imagine an operation whereby we "cut" the force of gravity between the body and the earth and patch it up again by a spring.
The unit of force used in engineering is the pound, which is the weight or gravity force of a standard piece of platinum at a specific location on earth. The weight of this standard piece varies slightly from place to place, being about 0.5 per cent greater near the North Pole than at the equator. This difference is too small to be considered for practical calculations in engineering statics, but it is sufficiently important for some effects in physics. (A method for exploring for oil deposits is based on these very slight variations in gravity from place to place.)
For the practical measurement of force, springs are often used. In a spring the elongation (or compression) is definitely related to the force on it so that a spring can be calibrated against the standard pound and then becomes a "dynamometer" or force meter.
A force is characterized not only by its magnitude but also by the direction in space in which it acts; it is a "vector" quantity, and not a "scalar" quantity. The line along which the force acts is called its "line of action." Thus, in order to specify a force completely, we have to specify its line of action and its magnitude. By making this magnitude positive or negative, we determine the direction of the force along the line of action.
A very important property of forces is expressed by the first axiom of statics, also known as Newton's third law (page 175), which states that action equals reaction. Contact forces are always exerted by one body on another body, and the axiom states that the force by the first body on the second one is equal and opposite to the force by the second body on the first one. For example, the push down on the table by a book is equal to the push up on the book by the table. Thinking about our imaginary experiment of inserting a thin "dynamometer" spring between the book and the table, the proposition looks to be quite obvious: there is only one force in the spring. Newton's third law, however, states that it is true not only for contact forces but also for gravity (and similar) forces. The earth pulls down on a flying airplane with a force equal to the one with which the airplane pulls up on the earth. This is less obvious, and in fact the proposition is of the nature of an axiom that cannot be proved by logical deduction from previous knowledge. It appeals to the intuition, and the logical deductions made from it (the entire theory of statics) conform well with experiment.
Another proposition about forces, which appeals to our intuition but which cannot be proved by logic, is the second axiom of statics, or the principle of transmissibility:
The state of equilibrium of a body is not changed when the point of action of a force is displaced to another point on its line of action. This means in practice that a force can be shifted along its own line without changing the state of equilibrium of a body. For ropes or struts in contact with the body, the proposition looks obvious; it should make no difference whether we pull on a short piece of rope close to the body or at the end of a long rope far away, provided that the short rope coincides in space with a piece of the longer one. For gravity forces the proposition is not so obvious.
It is noted that the equivalence of two forces acting at different points along their own line extends only to the state of equilibrium of a body, not to other properties. For instance, the stress in the body is definitely changed by the location of the point of action. Imagine a bar of considerable weight located vertically in space. Let the bar in case a be supported by a rope from the top and in case b, by a bearing at the bottom. The supporting force in both cases equals the weight and is directed upward along the bar; in case a it acts at the top, and in case b, at the bottom. By the axiom this should not make any difference in the state of equilibrium of the body, but in case a the bar is in tension, and in case b, in compression.
3. Parallelogram of Forces. The statements of action equals reaction and of transmissibility are not the only axioms about forces which are made. The third axiom in statics is that of the parallelogram of forces:
If on a body two forces are acting, whose lines of action intersect, then the equilibrium of the body is not changed by replacing these two forces by a single force whose vector is the diagonal of the parallelogram constructed on the two original forces.
This is illustrated in Fig. 1. The two forces, F1 and F2, have lines of action intersecting at O; they are said to be concurrent forces, to distinguish them from forces of which the lines of action do not intersect. The single force R, called the resultant, is equivalent to the combined action of the two forces, F1 and F2. Although this construction is now familiar to almost everyone, it is emphasized that it is an axiom, not provable by logic from known facts. It is based on experiment only, and the ancient Greeks and Romans did not know it, although Archimedes was familiar with the equilibrium of levers. The statement is about 350 years old and was formulated less than a century before the great days of Newton.
Many simple experiments can be devised to verify the axiom of the parallelogram of forces. They all employ for their interpretation two more statements which are sometimes also called axioms but which are so fundamental that they hardly deserve the honor. They are
Fourth axiom: A body in equilibrium remains in equilibrium when no forces are acting on it.
Fifth axiom: Two forces having the same line of action and having equal and opposite magnitudes cancel each other.
For an experimental verification of the parallelogram law, arrange the apparatus of Fig. 2, consisting of two freely rotating frictionless pulleys, P1 and P2, of which the axles are rigidly mounted, and a completely flexible string or rope strung over them. Three different weights, W1, W2, and W3 are hung on the string, and if the string is so long that the weights are kept clear of the pulleys, they will find a position of equilibrium, as the experiment shows. We observe the geometry of this position and reason by means of the five axioms. In this reasoning we employ a device, called isolation of the body, that is used in practically every problem in mechanics and that is of utmost importance. The first "body" we "isolate" consists of the weight W1 and a short piece of its vertical string attached to it. The "isolation" is performed by making an imaginary cut in the string just above W1 and by considering only what is below that cut. We observe from the experiment that W1 is in equilibrium and notice that two forces are acting on it: the pull of the string up and the weight W1 down. The lines of action of the two forces are the same so that by the fourth and fifth axioms combined, we deduce that the force or tension in the string is equal to the weight W1. Next we look at or "isolate" the pulley P1. The tensions in the two sections of string over a frictionless pulley are the same so that the tension in the string between P1 and A is still W1. This is not obvious, and the proof of it will be given much later, on page 21.
By entirely similar reasoning we conclude from the fourth and fifth axioms that the tension in the string between P2 and A is W2 and that the tension in the vertical piece of string between A and W3 is equal to W3.
Now we once more isolate a body, and choose for it the knot A and three short pieces of string emanating from it. This body is in equilibrium by experiment, and we notice that there are three forces acting on it, the string tensions W1, W2, and W3, whose lines of action intersect at A. Now by the third axiom of the parallelogram of forces, the tensions F1 and F2 of Fig. 2 (which we have seen are equal to W1 and W2) add up to the resultant R. By the fourth and fifth axioms, it is concluded that R must be equal (and opposite) to W3. By hanging various weights on the strings we can form parallelograms of all sorts of shapes and so verify the third axiom experimentally.
For example, if W1 = W2 = 1 lb and , then the angles of the parallelogram will be 45 and 90 deg.
In constructing the parallelogram of forces, not all the lines have to be drawn in the figure. In Fig. 1 it is seen that the distance F2R is equal (and parallel) to OF1. In order to find the resultant OR in Fig. 1, it suffices to lay off the force OF2, and then starting at the end point F2 to lay off the other force F2R = OF1. The resultant OR is then the closing line of the triangle OF2R. The lines OF1 and F1R do not necessarily have to be drawn. The construction then is called the triangle of forces.
If more than two forces are to be added together, this triangle construction leads to a much simpler figure than the parallelogram construction.
Excerpted from Mechanics by J. P. Den Hartog. Copyright © 1948 J. P. Den Hartog. Excerpted by permission of Dover Publications, Inc..
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