The golden rule of mechanics

Teachers begin a physics course in general education schools with an acquaintance with an important section of this science – mechanics. Isaac Newton made a great contribution to its development. However, humanity knows other philosophers and scientists who have no less important merits.

The moment of force and the rule of mechanic

To understand the whole physical essence of the golden law of mechanics, it is necessary to get acquainted with such an important concept as the moment of force. Let there be some system that consists of the following elements:

Material point P, which is rigidly connected to the O axis. The distance from O to P is the length of the segment OP. Let some force F be applied to the point P perpendicular to the axis O. Then the moment M of this force will be equal to:

M = [OP * F].

The moment of force is measured in newtons per meter (N * m) in the SI system. Note that these units correspond to work, measured in joules (J). This fact is not accidental, since the product of the moment M by the angle by which the object rotates around the axis is the work performed by the force F.

The entered physical quantity is not exclusively theoretical, it has a direct practical significance. The essence of the moment of force is that it shows the possibility or ability of the force to rotate in the system under consideration. For example, if you push a door near the handle, then you can open it even with the efforts of one little finger. On the contrary, it is necessary to use the force of the whole hand to open the door, pushing it near the hinges.

Simple mechanisms

Since ancient times, mankind has been using improvised means to facilitate physical labor. In modern science, it is customary to call simple mechanisms such objects that allow transforming the magnitude and direction of external efforts, while they themselves are not energy generators. This class of mechanisms includes the following:

• lever arm;
• block;
• inclined plane;
• gate;
• wedge;
• screw;
• wheel.

Some physicists continue to argue about the minimum number of simple mechanisms and the possibility of presenting some of them as a combination of others. Each of them can simultaneously transform the direction of the applied force and its magnitude. For example, when tightening a nut, a person spends little effort performing a separate turn. These forces are converted, firstly, into a significant generated pressure, and secondly, its direction changes from perpendicular to the axis of the nut or bolt to parallel to it.

There are two types of levers – a lever of the first and a lever of the second kind. A lever of the first kind is a lever whose axis of rotation is located between the points of application of forces, and the forces themselves are directed in one direction. A lever of the second kind is a lever whose axis of rotation is located on one side of the points of application of forces, and the forces themselves are directed opposite to each other.

Historical reference

According to generally accepted concepts, simple mechanisms were used by the ancient Babylonians and Egyptians during the creation of their structures of stone. However, only the works of the Greek ancient philosophers on the issues of mechanics have survived to this day in writing. Two well-known surnames should be distinguished:

• Heron of Alexandria;
• Archimedes.

It was these two philosophers who made the main contribution to the development of the physical theory of simple mechanisms. Thus, Archimedes, who lived in Greece in the 3rd century BC, established the rule of the lever. According to one of the legends, he was able to use it in practice, when alone, with the help of a system of blocks and levers, which he reinforced with ropes, he was able to lift a ship loaded with passengers and goods above the water surface.

It is known about Heron that he lived in the 1st century AD, that is, 3 centuries after Archimedes, in the Egyptian city of Alexandria. He is known as the greatest inventor during the heyday of Hellenic culture and science.

For the first time from the point of view of mathematics, Geron considered all simple mechanisms and generalized the results obtained in the form of the golden rule of mechanics.

Law of energy conservation

The golden rule of mechanics says: when using absolutely any simple mechanism, how many times there is a gain in strength, the same amount of times there is a loss in movement, and vice versa.

By this definition, we can safely say that it is nothing more than the law of conservation of energy. Let there be a mechanism that is capable, as a result of the action of an external force F1 applied to an arbitrary point O1, to convert this force into the value F2, which will already be applied to the point O2. During the operation of a simple mechanism, both points will move to distances d1 and d2, respectively. Then the golden law can be written in the following form:

F1 * d1 = F2 * d2.

The product of force on the path gives work. Thus, this equality shows the comparability of the work performed by two forces, that is, it speaks of the law of conservation of energy.

Then two more possible expressions are obtained:

F1 * v1 = F2 * v2;

N1 = N2.

Here v1, v2 are the speeds of movement of points O1 and O2, respectively. N1 and N2 – powers developed by forces F1 and F2.

Archimedes lever

A good help in understanding the golden law of mechanics is a presentation of how a simple lever works. It consists of a fulcrum and either one or two shoulders, depending on the gender. There are three kinds of this mechanism:

• The point of application of the force and the point of resistance to it are on opposite sides of the support.  Examples of levers of the first kind are scissors, pliers, catapult, action of human joints and muscles.
• External force and resistance act on one side of the support so that the resistance shoulder is always less than the external force shoulder. This arrangement of the acting forces makes it possible to always win in strength and lose in movement. Examples of second class lever mechanisms are the one-wheeled hand wheelbarrow and the nutcracker.
• If an external force and resistance are applied on one side of the support, but the first lies closer to it than the second, then we are talking about a lever of the third kind. The point of using it is to win. in movement, although this requires the sacrifice of the effort involved. Examples include fishing rod and eyebrow tweezers.

For levers of all kinds, the following formula is valid, which was received by Archimedes:

F1 * d1 = F2 * d2.

The product of the force by the length of the arm is a constant value.

This expression shows equality of moments. Indeed, according to the definition, the product F * d can be considered as the moment of force F. The golden rule of the lever in this case is transformed into the following form:

M1 = M2.

Equality of moments is one of two conditions of equilibrium in systems with axes of rotation. The second condition is the equality to zero of all forces acting in the system. In the case of a lever that is in equilibrium, this condition can be written as follows:

F1 + F2 = N.

Where N is the reaction force of the support, the vector of which is directed in the direction opposite to the forces of action and reaction.

Example task

There is a lever of the first kind. At what distance should a weight of 1 kg be put in order for it to balance the ball, the mass of which is 10 kg and which is at a distance of 30 cm from the fulcrum.

To get the answer to the problem, a simple calculation should be performed according to the golden rule of leverage:

Students solve problems

F1 * d1 = F2 * d2 ==>

d1 = F2 * d2 / F1 = m2 * g * d2 / (m1 * g) = m2 * d2 / m1.

Converting the distance d2 = 30 cm into meters, and substituting the known masses, you can get the answer:

d1 = 10 * 0.3 / 1 = 3 meters.

Thus, the golden law of mechanics is universal.