Equilibrium of a mechanical system and the method of virtual displacements

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Physics 8, 9, 10

The note “Equilibrium of a mechanical system and the method of virtual displacements” published below is intended for eight-year-olds, the note “How many states does a substance have?” - for ninth-graders, “A few changes regarding the photo effect” - for tenth-graders.

Equilibrium of a mechanical system and the method of virtual displacements

He said that for the construction of one of the cathedrals in Switzerland, its architect came across blocks that allowed him to lift particularly heavy loads to great heights. He designed a very complex chain hoist, but got confused in the numerous tension forces of the cables and was never able to calculate how many people would need to be hired to service it. For help, the architect approached the famous scientist Juhann Bernoulli (1667-1748). Imagine the architect’s surprise when Bernoulli, barely looking at the drawing, immediately gave the answer. The amazed architect asked to reveal to him the secret of the pillar calculation…

Let us also try (following Bernoulli) to discuss the equilibrium conditions of a mechanical system from a slightly different position than was done in the next chapter of the school textbook “Physics 8”.

Obviously, in order to find out the conditions of equilibrium of a system, the reaction forces of mechanical connections are driven *). Connections are usually called those restrictions that are imposed on the position of individual parts of the system, or on the possibility of their movement.

*) See the article by A. I. Chernoutsan “Kinematic connections in diamnic problems” in the second issue of the journal for 1988.

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Examples of mechanical connections include a thread on which a load is suspended, a hinge connecting individual parts of a mechanism, a plane on which the body of the system is located, etc. The more connections there are in a system (for example, threads in a pulley block designed by our architect), the more difficult it is to monitor the action of all the reaction forces arising in them.

In many cases, mechanical connections have some remarkable property, which Bernoulli used as the basis for his simple and elegant method of finding the equilibrium conditions of a mechanical system. This property lies in the fact that the total work of all reaction forces arising in the connection of the system for any sufficiently small possible deviations from the equilibrium position is equal to zero.

Here it is necessary to specifically stipulate what is meant by “any sufficiently small possible deviations” of the system from the equilibrium position. First of all, of course, the movements of the points of the system that determine such deviations should not contradict the mechanical connections (the threads should not fray, the hinges should not break). Such movements are called emergent, or virtual (which means mental). They should be distinguished from true movements that occur under the influence of applied forces. Virtual displacements are not relative to the process of motion of the system - they are introduced only in order to identify the correspondence of forces existing in the system and establish equilibrium conditions. A small amount of displacement is needed so that the reaction forces can be considered unchanged.

Connections for which the sum of the elementary work of the reaction forces for any virtual displacements is equal to zero are called ideal. We emphasize that not all connections are ideal. So, if the bodies of the system lie on a plane with friction, then this connection (plane) is no longer ideal.

Let us consider how the property of ideal connections is fulfilled using the example of a system of points forming an absolutely rigid body. Any point $A$ of such a body is connected to all other points by reaction forces. Let us select some point $B$ and denote the reaction forces acting between the points by $\vec{F}_{AB}$ and $\vec{F}_{BA}$ (the first of them is applied to point $A$, the second - to $B$). Let’s forget for a moment about all other points of the body, except for the two highlighted ones. We are then left with a system of two point materials connected by an incompatible rigid rod. According to Newton’s third law,

$$\vec{F}_{AB} = -\vec{F}_{BA}.$$

Let us now perform some virtual movements of these points $\Delta s_A$ and $\Delta s_B$. We can assume that $\Delta s_A$ is the translational displacement of the rigid body as a whole, common to all points, and $\Delta s_B = \Delta s_A + \Delta s'$, where $\Delta s'$ is the displacement corresponding to the expression of point $B$ relative to point $A$. The total work done by the reaction forces during these virtual displacements is

$$\Delta A = (\vec{F}_{AB} \cdot \Delta s_A) + (\vec{F}_{BA} \cdot \Delta s_B) = \left( (\vec{F}_{AB} + \vec{F}_{BA}) \cdot \Delta s_A \right) + (\vec{F}_{BA} \cdot \Delta s').$$

The first term on the right side of this equality is equal to zero due to condition (*). The second term also turns out to be equal to zero, since the displacement $\Delta s'$, corresponding to the expression of the point $B$ relative to the point $A$, is perpendicular to the force $\vec{F}_{BA}$, which is directed along the straight line connecting the points $A$ and $B$. In the same way, it can be shown that the virtual work (that is, the total work during virtual displacements) of the reaction forces for any other rigid body is equal to zero. Thus, by considering the example, we were convinced of the validity of the discussed property of ideal connections.

In order to finally formulate a new principle of equilibrium, it remains for us to consider the virtual work of external forces. Note that each external force applied to the system under equilibrium conditions must be in equilibrium with the reaction forces caused by