Conditions for the equilibrium of a solid. Conditions for the equilibrium of bodies

Statics.

A branch of mechanics that studies the conditions for the equilibrium of mechanical systems under the action of forces and moments applied to them.

The balance of power.

Mechanical balance, also known as static equilibrium, is the state of a body at rest, or moving uniformly, in which the sum of the forces and moments acting on it is zero

Equilibrium conditions for a rigid body.

The necessary and sufficient conditions for the equilibrium of a free rigid body are the equality to zero of the vector sum of all external forces acting on the body, the equality to zero of the sum of all moments of external forces about an arbitrary axis, the equality to zero of the initial velocity of the translational motion of the body, and the condition of equality to zero of the initial angular velocity of rotation.

Types of balance.

Body balance is stable if, for any small deviations from the equilibrium position allowed by external constraints, forces or moments of forces arise in the system, tending to return the body to its original state.

The balance of the body is unstable, if at least for some arbitrarily small deviations from the equilibrium position allowed by external constraints, forces or moments of forces arise in the system that tend to deviate the body even more from the initial state of equilibrium.

The balance of the body is called indifferent, if for any small deviations from the equilibrium position allowed by external constraints, forces or moments of forces arise in the system, tending to return the body to its original state

Center of gravity of a rigid body.

center of gravity body is called the point, relative to which the total moment of gravity acting on the system is equal to zero. For example, in a system consisting of two identical masses connected by an inflexible rod and placed in an inhomogeneous gravitational field (for example, planets), the center of mass will be in the middle of the rod, while the center of gravity of the system will be shifted to that end of the rod, which is closer to the planet (because the weight of the mass P = m g depends on the gravitational field parameter g), and, generally speaking, is even located outside the rod.

In a constant parallel (homogeneous) gravitational field, the center of gravity always coincides with the center of mass. Therefore, in practice, these two centers almost coincide (since the external gravitational field in non-space problems can be considered constant within the volume of the body).

For the same reason, the concepts of center of mass and center of gravity coincide when these terms are used in geometry, statics, and similar areas, where its application in comparison with physics can be called metaphorical and where the situation of their equivalence is implicitly assumed (since there is no real gravitational field and it makes sense to take into account its heterogeneity). In these uses, the two terms are traditionally synonymous, and often the second is preferred simply because it is older.

Statics is a branch of mechanics that studies the balance of bodies. Statics allows you to determine the conditions for the equilibrium of bodies and answers some questions that relate to the movement of bodies, for example, gives an answer in which direction the movement occurs if the balance is disturbed. It is worth looking around and you will notice that most bodies are in equilibrium - they are either moving at a constant speed or at rest. This conclusion can be drawn from Newton's laws.

An example is the person himself, a picture hanging on the wall, cranes, various buildings: bridges, arches, towers, buildings. The bodies around us are exposed to some kind of force. A different number of forces acts on bodies, but if we find the resulting force, for a body in equilibrium, it will be equal to zero.
Distinguish:

static equilibrium - the body is at rest;
dynamic equilibrium - the body moves at a constant speed.

static balance. If forces F1, F2, F3, and so on, act on the body, then the main requirement for the existence of an equilibrium state is (equilibrium). This is a vector equation in 3D space, and represents three separate equations, one for each direction in space. .

The projections of all forces applied to the body in any direction must be compensated, that is, the algebraic sum of the projections of all forces in any direction must be equal to 0.

When finding the resultant force, you can transfer all the forces and place the point of their application at the center of mass. The center of mass is a point that is introduced to characterize the movement of a body or a system of particles as a whole, characterizes the distribution of masses in the body.

In practice, we very often encounter cases of both translational and rotational movement at the same time: a barrel rolling down an inclined plane, a dancing couple. With such a movement, one equilibrium condition is not enough.

The necessary equilibrium condition in this case will be:

In practice and in life plays an important role body stability characterizing the balance.

There are types of balance:

Stable balance;
Unstable equilibrium;
Indifferent balance.

sustainable balance- this is equilibrium, when, with a small deviation from the equilibrium position, a force arises that returns it to a state of equilibrium (a pendulum of a stopped clock, a tennis ball rolled into a hole, a roly-poly or tumbler, linen on a rope are in a state of stable equilibrium).

Unstable equilibrium- this is a state when the body, after being removed from the equilibrium position, deviates even more from the equilibrium position due to the emerging force (tennis ball on a convex surface).

Indifferent balance- being left to itself, the body does not change its position after being removed from the state of equilibrium (a tennis ball lying on the table, a picture on the wall, scissors, a ruler suspended on a carnation are in a state of indifferent equilibrium). The axis of rotation and the center of gravity are the same.

For two bodies, the body will be more stable, which has larger footprint.

A body is at rest (or moves uniformly and in a straight line) if the vector sum of all forces acting on it is zero. The forces are said to balance each other. When we are dealing with a body of a certain geometric shape, when calculating the resultant force, all forces can be applied to the center of mass of the body.

The condition for the equilibrium of bodies

In order for a body that does not rotate to be in equilibrium, it is necessary that the resultant of all forces acting on it be equal to zero.

F → = F 1 → + F 2 → + . . + F n → = 0 .

The figure above shows the equilibrium of a rigid body. The block is in a state of equilibrium under the action of three forces acting on it. The lines of action of the forces F 1 → and F 2 → intersect at the point O. The point of application of gravity is the center of mass of the body C. These points lie on one straight line, and when calculating the resultant force F 1 → , F 2 → and m g → are reduced to point C .

The condition that the resultant of all forces be equal to zero is not enough if the body can rotate around some axis.

The shoulder of the force d is the length of the perpendicular drawn from the line of action of the force to the point of its application. The moment of force M is the product of the arm of the force and its modulus.

The moment of force tends to rotate the body around its axis. Those moments that rotate the body counterclockwise are considered positive. The unit of measurement of the moment of force in the international SI system is 1 Newton meter.

Definition. moment rule

If the algebraic sum of all the moments applied to the body relative to the fixed axis of rotation is equal to zero, then the body is in equilibrium.

M1 + M2 + . . + M n = 0

Important!

In the general case, for the equilibrium of bodies, two conditions must be met: the resultant force is equal to zero and the rule of moments is observed.

There are different types of equilibrium in mechanics. Thus, a distinction is made between stable and unstable, as well as indifferent equilibrium.

A typical example of an indifferent equilibrium is a rolling wheel (or ball), which, if stopped at any point, will be in a state of equilibrium.

Stable equilibrium is such an equilibrium of a body when, with its small deviations, forces or moments of forces arise that tend to return the body to an equilibrium state.

Unstable equilibrium - a state of equilibrium, with a small deviation from which the forces and moments of forces tend to bring the body out of balance even more.

In the figure above, the position of the ball is (1) - indifferent equilibrium, (2) - unstable equilibrium, (3) - stable equilibrium.

A body with a fixed axis of rotation can be in any of the described equilibrium positions. If the axis of rotation passes through the center of mass, there is an indifferent equilibrium. In stable and unstable equilibrium, the center of mass is located on a vertical line that passes through the axis of rotation. When the center of mass is below the axis of rotation, the equilibrium is stable. Otherwise, vice versa.

A special case of equilibrium is the equilibrium of a body on a support. In this case, the elastic force is distributed over the entire base of the body, and does not pass through one point. A body is at rest in equilibrium when a vertical line drawn through the center of mass intersects the area of support. Otherwise, if the line from the center of mass does not fall into the contour formed by the lines connecting the support points, the body overturns.

An example of the balance of a body on a support is the famous Leaning Tower of Pisa. According to legend, Galileo Galilei dropped balls from it when he conducted his experiments on the study of the free fall of bodies.

A line drawn from the center of mass of the tower intersects the base approximately 2.3 m from its center.

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Equilibrium of a mechanical system is a state in which all points of a mechanical system are at rest with respect to the reference frame under consideration. If the frame of reference is inertial, the equilibrium is called absolute, if non-inertial — relative.

To find the equilibrium conditions for an absolutely rigid body, it is necessary to mentally divide it into a large number of sufficiently small elements, each of which can be represented by a material point. All these elements interact with each other - these interaction forces are called internal. In addition, external forces can act on a number of points of the body.

According to Newton's second law, for the acceleration of a point to be zero (and the acceleration of a point at rest to be zero), the geometric sum of the forces acting on that point must be zero. If the body is at rest, then all its points (elements) are also at rest. Therefore, for any point of the body, we can write:

where is the geometric sum of all external and internal forces acting on i th element of the body.

The equation means that for the equilibrium of a body it is necessary and sufficient that the geometric sum of all forces acting on any element of this body is equal to zero.

From it is easy to obtain the first condition for the equilibrium of a body (system of bodies). To do this, it is enough to sum the equation over all elements of the body:

The second sum is equal to zero according to Newton's third law: the vector sum of all internal forces of the system is equal to zero, since any internal force corresponds to a force equal in absolute value and opposite in direction.

Hence,

The first condition for the equilibrium of a rigid body(body systems) is the equality to zero of the geometric sum of all external forces applied to the body.

This condition is necessary but not sufficient. It is easy to verify this by remembering the rotating action of a pair of forces, the geometric sum of which is also equal to zero.

The second condition for the equilibrium of a rigid body is the equality to zero of the sum of the moments of all external forces acting on the body, relative to any axis.

Thus, the equilibrium conditions for a rigid body in the case of an arbitrary number of external forces look like this: