The Different Principles Of Statics

Statics is the science of the equilibrium of forces.

By force or power is the cause which impresses or tends to impress motion on the body to which it is supposed to be applied.

It is also by the quantity of motion impressed, or about to be impressed, that force or power must be estimated.

In the state of equilibrium, force has no actual exercise.

• It produces only a simple tendency to motion.

But it must always be measured by the effect it would produce if it were not stopped.

By taking any force or its effect as unity, the expression of any other force is only a ratio, a mathematical quantity, represented by numbers or lines.

It is from this point of view that forces must be considered in Mechanics.

Equilibrium results from the destruction of several forces which oppose each other and which mutually annihilate the action they exert on one another.

Statics aims to give the laws on how this destruction operates.

These laws are founded on 3 general principles:

1. The lever
2. The composition of forces
3. Virtual velocities.

The Lever

1. Archimedes was the only one among the ancients who has left us a theory of equilibrium in his two Books On the Equilibrium of Planes or On the Equilibrium of Planes.

He is the author of the principle of the lever: if a straight lever is loaded with 2 weights on either side of the fulcrum, at distances from this point reciprocally proportional to the same weights:

• this lever will be in equilibrium
• its support will be loaded with the sum of the 2 weights.

Archimedes takes this principle as a self-evident axiom of Mechanics, or at least as a principle of experience.

He reduces to this simple and primitive case that of unequal weights, by imagining these weights, when they are commensurable, divided into several parts all equal to each other, and by supposing that the parts of each weight are separated and transported, on either side, on the same lever, at equal distances, so that the lever is loaded with several small equal weights placed at equal distances around the fulcrum.

Then he demonstrates the truth of the same theorem for incommensurable weights, by means of the method of exhaustion, by showing that there could be no equilibrium between these weights, unless they are in inverse ratio to their distances from the fulcrum.

Some modern authors, such as Stevin in his Statics, and Galileo in his Dialogues on Motion, have made Archimedes’ demonstration simpler.

They supposed that the weights attached to the lever are 2 horizontal parallelepipeds suspended by their middle, and whose widths and heights are equal, but whose lengths are double the lever arms corresponding to them inversely.

In this way, the 2 parallelepipeds are in inverse ratio to their lever arms.

At the same time, they are placed end to end, so that they form only a single one, whose midpoint corresponds precisely to the fulcrum of the lever.

Archimedes had already used a similar consideration to determine the center of gravity of a magnitude composed of two parabolic surfaces, in the first proposition of the second Book On the Equilibrium of Planes.

Other authors, on the contrary, have thought they found defects in Archimedes’ demonstration.

They have turned it in different ways to make it more rigorous.

Huygens supplemented Archimedes’ demonstration with Demonstratio æquilibrii bilancis[1] in 1693.

He observes that Archimedes tacitly supposes that if several equal weights are applied to a horizontal lever at equal distances, they exert the same force to incline the lever.

This is whether:

• they are all on the same side of the fulcrum, or
• some are on one side and others on the other side of the fulcrum

Archimedes distributes the aliquot parts of the 2 commensurable weights on the same lever on either side of the points where the whole weights are supposed to be applied.

But to avoid this precarious supposition, Huygens distributes them in the same way, but on two other horizontal levers, placed perpendicularly at the ends of the main lever, in the form of a T.

In this way, one has a horizontal plane loaded with several equal weights, and which is evidently in equilibrium on the line of the first lever, because the weights are distributed equally and symmetrically on both sides of this line.

But Huygens demonstrates that this plane is also in equilibrium on a line inclined to that one, and passing through the point which divides the primitive lever into parts reciprocally proportional to the weights with which it is supposed to be loaded, because he shows that the small weights are also placed at equal distances on both sides of the same line; from which he concludes that the plane, and consequently the proposed lever, must be in equilibrium on the same point.

This demonstration is ingenious.

But it does not entirely supply what one might indeed desire in that of Archimedes.

2. The equilibrium of a straight and horizontal lever, whose ends are loaded with equal weights, and whose fulcrum is at the middle of the lever, is a self-evident truth.

This is because there is no reason why one of the weights should prevail over the other, everything being equal on both sides of the fulcrum.

It is not the same with the supposition that the load on the support is equal to the sum of the two weights.

All mechanicians have taken this as a result of daily experience, which teaches that the weight of a body depends only on its total mass, and not at all on its shape[2].

One can nevertheless deduce this truth from the first, by considering, like Huygens, the equilibrium of a plane on a line.

For this, one need only imagine a triangular plane loaded with two equal weights at the two ends of its base, and with a double weight at its apex.

This plane will evidently be in equilibrium, being supported on a straight line or fixed axis, which passes through the middle of the two sides of the triangle; for one can regard each of these sides as a lever loaded at its two ends with two equal weights, and which has its fulcrum on the axis which passes through its middle.

This equilibrium can be considered in another way, by regarding the base itself of the triangle as a lever whose ends are loaded with two equal weights, and by imagining a transverse lever which joins the apex of the triangle and the middle of its base in the form of a T, one of whose ends is loaded with the double weight placed at the apex, and the other serves as a fulcrum for the lever forming the base.

This last lever will be in equilibrium on the transverse lever which supports it at its middle, and that the latter will, consequently, be in equilibrium on the axis on which the plane is already in equilibrium.

As the axis passes through the middle of the two sides of the triangle, it will also necessarily pass through the middle of the line drawn from the apex of the triangle to the middle of its base.

Therefore, the transverse lever will have its fulcrum at the midpoint and must, consequently, be equally loaded at both ends; therefore the load supported by the fulcrum of the lever which forms the base of the triangle, and which is loaded at its two ends with equal weights, will be equal to the double weight of the apex and, consequently, equal to the sum of the two weights.

If, instead of a triangle, one considered a trapezoid loaded at its four corners with 4 equal weights, one would find in the same way that the two levers of unequal lengths, forming the parallel sides of the trapezoid, exert on their fulcrums equal forces.

3. Archimedes substitutes for a weight in equilibrium on a lever two weights each equal to half of this weight and placed on the same lever, at equal distances on either side of the point where the weight is attached.

For the action of this weight is the same as that of a lever suspended by its middle at the same point and loaded, at its two ends, with two weights each equal to half of the same weight.

Nothing prevents approaching this last lever to the first, so that it becomes part of it.

Or else, which is perhaps more rigorous, one need only regard this last lever as being held in equilibrium by a force applied at its midpoint, directed from bottom to top, and equal to the weight whose two halves are supposed to be applied at its ends.

Then, by applying this lever in equilibrium on the first lever which is supposed to be in equilibrium on its fulcrum, the total equilibrium will always subsist, and, if the application is made in such a way that the middle of the second lever coincides with the end of one of the arms of the first lever, the force which sustains the second lever can be supposed to be applied to the very weight with which this arm is loaded, and which, being supported, will have no more action on the lever, but will thus be replaced by two weights each equal to half of it and placed on either side of this weight on the extended first lever. This superposition of equilibria is, in Mechanics, a principle as fruitful as is, in Geometry, the superposition of figures.

4. One can therefore regard the equilibrium of a straight and horizontal lever, loaded with two weights in inverse ratio to their distances from the fulcrum of the lever, as a rigorously demonstrated truth.

By the principle of superposition, it is easy to extend it to any angular lever, whose fulcrum would be at the angle and whose arms would be pulled in opposite directions by forces perpendicular to their directions.

An angular lever with equal arms, and movable around the vertex of the angle, will be held in equilibrium by two equal forces applied perpendicularly to the ends of the two arms, and tending to make them turn in opposite directions.

If, therefore, one has a straight lever in equilibrium, one of whose arms is equal to those of the angular lever and is loaded at its end with a weight equivalent to each of the powers applied to the angular lever, the other arm being loaded with the weight necessary for equilibrium, and one superposes these levers so that the vertex of the angle of one falls on the fulcrum of the other, and that the equal arms of one and the other coincide and form only one, the power applied to the arm of the angular lever will sustain the weight suspended from the equal arm of the straight lever, so that one can disregard both, and suppose the arm formed by the union of these two annihilated.

Equilibrium will therefore still subsist between the other two arms forming an angular lever pulled at its ends by perpendicular forces and in inverse ratio to the length of the arms, as in the straight lever.

Now a force can be supposed to be applied to any point whatsoever of its direction. Therefore two forces, applied to any points of a plane held by a fixed point and directed as one wishes in this plane, are in equilibrium when they are in inverse ratio to the perpendiculars drawn from this point to their directions; for one can regard these perpendiculars as forming an angular lever whose fulcrum is the fixed point of the plane; this is what is now called the principle of moments, meaning by moment the product of a force by the lever arm through which it acts.

This general principle suffices to solve all problems of Statics.

The consideration of the winch had made it perceived from the first steps taken after Archimedes, in the theory of simple machines, as can be seen in the Work of Guido Ubaldo, entitled Mecanicorum liber, which appeared in Pesaro, in 1577.

But he did not know how to apply it to the inclined plane, nor to other machines which depend on it, like the wedge and the screw of which he gave only a rather inaccurate theory.