Statics is the branch of mechanics that studies the conditions of equilibrium of bodies.
From Newton's second law it follows that if the geometric sum of all external forces, applied to the body, is equal to zero, then the body is at rest or performs a uniform straight motion. In this case, it is customary to say that the forces applied to the body balance each other. When calculating resultant all forces acting on a body can be applied to center of mass .
For a non-rotating body to be in equilibrium, it is necessary that the resultant of all forces applied to the body be equal to zero.
In Fig. 1.14.1 gives an example of equilibrium solid under the influence of three forces. Intersection point O lines of action of forces and does not coincide with the point of application of gravity (center of mass C), but in equilibrium these points are necessarily on the same vertical. When calculating the resultant, all forces are reduced to one point.
If the body can rotate relative to some axis, then for its equilibrium It is not enough for the resultant of all forces to be zero.
The rotating effect of a force depends not only on its magnitude, but also on the distance between the line of action of the force and the axis of rotation.
The length of the perpendicular drawn from the axis of rotation to the line of action of the force is called shoulder of strength.
Product of the modulus of force per arm d called moment of force M. The moments of those forces that tend to turn the body counterclockwise are considered positive (Fig. 1.14.2).
Rule of Moments : a body having a fixed axis of rotation is in equilibrium if algebraic sum the moments of all forces applied to the body relative to this axis are equal to zero:
In the International System of Units (SI), moments of forces are measured in NNewton— meters (N∙m) .
IN general case, when a body can move translationally and rotate, for equilibrium it is necessary to fulfill both conditions: the resultant force being equal to zero and the sum of all moments of forces being equal to zero.
Rolling on horizontal surface wheel - example indifferent equilibrium(Fig. 1.14.3). If the wheel is stopped at any point, it will end up in equilibrium state. Along with indifferent equilibrium, mechanics distinguishes between states sustainable And unstable balance.
A state of equilibrium is called stable if, with small deviations of the body from this state, forces or moments of force arise that tend to return the body to an equilibrium state.
With a small deviation of the body from a state of unstable equilibrium, forces or moments of force arise that tend to remove the body from the equilibrium position.
A ball lying on a flat horizontal surface is in a state of indifferent equilibrium. A ball located at the top of a spherical protrusion is an example of unstable equilibrium. Finally, the ball at the bottom of the spherical recess is in a state of stable equilibrium (Fig. 1.14.4).
For a body with a fixed axis of rotation, all three types of equilibrium are possible. Indifference equilibrium occurs when the axis of rotation passes through the center of mass. With stable and not stable equilibrium the center of mass is on a vertical straight line passing through the axis of rotation. Moreover, if the center of mass is below the axis of rotation, the state of equilibrium turns out to be stable. If the center of mass is located above the axis, the state of equilibrium is unstable (Fig. 1.14.5).
A special case is the balance of a body on a support. In this case elastic force the support is not applied to one point, but is distributed along the base of the body. A body is in equilibrium if vertical line, drawn through the center of mass of the body, passes through support area, i.e. inside the contour formed by lines connecting the support points. If this line does not intersect the area of support, then the body tips over. An interesting example the balance of a body on a support is the leaning tower in the Italian city of Pisa (Fig. 1.14.6), which, according to legend, was used by Galileo when studying the laws free fall tel. The tower has the shape of a cylinder with a height of 55 m and a radius of 7 m. The top of the tower is deviated from the vertical by 4.5 m.
A vertical line drawn through the center of mass of the tower intersects the base approximately 2.3 m from its center. Thus, the tower is in a state of equilibrium. The balance will be broken and the tower will fall when the deviation of its top from the vertical reaches 14 m. Apparently, this will not happen very soon.
BODY EQUILIBRIUM
“Give me a foothold and I will lift the Earth.”
Archimedes
Equilibrium conditions.
- I equilibrium condition:
- A body is in equilibrium if the geometric sum of external forces applied to the body is equal to zero.
∑ F=0.
- II equilibrium condition:
- The sum of the moments of forces acting clockwise must be equal to the sum of the moments of forces acting counterclockwise.
∑ M per hour. =∑ M against the hour.
- М = F l, where М – moment of force, F – force, l – arm of force – shortest distance from the fulcrum to the line of action of the force.
Center of gravity of the body.
- The center of gravity of a body is the point through which the resultant of all parallel forces gravity acting on individual elements bodies.
- Find the center of gravity of these figures.
- Find the center of gravity of these figures.
- Find the center of gravity of these figures.
- Find the center of gravity of these figures.
TYPES OF EQUILIBRIUM
Indifferent
Sustainable
Unstable
If balancing forces act on a supported body, then the body is in the position balance.
When a body deviates from its equilibrium position, the balance of forces is also disrupted. If a body returns to its original position under the action of a resultant force, then this is - stable equilibrium .
If the body, under the action of the resultant force, deviates even more from the equilibrium position, then this is unstable equilibrium .
It is possible that in any position of the body, the balance of forces is maintained. This condition is called indifferent balance .
Conclusion :
- Equilibrium is stable if, with a small deviation from the equilibrium position, there is a force tending to return it to this position.
- A stable position is in which it potential energy minimal.
If the center of gravity is located below the fulcrum, the balance of the body or system of bodies is sustainable . When the body deviates, the center of gravity rises and the body returns to its original state.
The balance of a body having a fulcrum below the center of gravity is unstable. But balance can restore by shifting the fulcrum of the body in the direction of shifting the center of gravity.
By the position of the center of gravity one can judge the type of equilibrium. For example, a tightrope walker riding a bicycle with a counterweight is an example stable equilibrium .
Conclusion :
- For the stability of a body located at one point or line of support, it is necessary that the center of gravity be below the point (line) of support.
If, when a body that has a support area deviates, the center of gravity increases, then the equilibrium will be stable. At stable equilibrium a vertical line passing through the center of gravity will always pass through the area of support.
Two bodies that have the same weight and support area, but different heights, have different limit angle tilt If this angle is exceeded, the bodies tip over.
At a lower center of gravity, it is necessary to spend great job to tip over the body. Therefore, the overturning work can serve as a measure of its stability.
Unstable equilibrium
Stable balance
Conclusion :
1. The body that has the largest support area is stable.
2. Of two bodies of the same area, the one whose center of gravity is lower is stable, because it can be tilted without tipping over at a large angle.
- There are three types of equilibrium: stable, unstable, indifferent.
- A stable position of a body in which its potential energy is minimal.
- Stability of bodies on flat surface the more than larger area supports and lower center of gravity.
A body is at rest (or moves uniformly and rectilinearly) if the vector sum of all forces acting on it is equal to zero. They say that forces balance each other. When we are dealing with a certain body geometric shape, when calculating the resultant force, all forces can be applied to the center of mass of the body.
Condition for equilibrium of bodies
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 influence of three forces acting on it. The lines of action of the forces F 1 → and F 2 → intersect at point O. The point of application of gravity is the center of mass of the body C. These points lie on the same straight line, and when calculating the resultant force F 1 →, F 2 → and m g → are brought to point C.
The condition that the resultant of all forces be equal to zero is not enough if the body can rotate around a certain axis.
The arm of 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 force arm and its modulus.
The moment of force tends to rotate the body around its axis. Those moments that turn the body counterclockwise are considered positive. Unit of measurement of moment of force in international system SI - 1 Newton meter.
Definition. Rule of Moments
If the algebraic sum of all moments applied to a body with respect to fixed axis rotation is zero, then the body is in a state of equilibrium.
M 1 + M 2 + . . +Mn=0
Important!
In the general case, for bodies to be in equilibrium, two conditions must be met: the resultant force must be equal to zero and the rule of moments must be observed.
In mechanics there is different types balance. Thus, there is a distinction between stable and unstable, as well as indifferent equilibrium.
A typical example of 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 is a state of equilibrium, with a small deviation from which forces and moments of forces tend to throw 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, indifference equilibrium occurs. With stable and unstable equilibrium the center of mass is located on a vertical straight line that passes through the axis of rotation. When the center of mass is below the axis of rotation, the equilibrium is stable. Otherwise, it's the other way around.
A special case of balance is the balance of a body on a support. In this case, the elastic force is distributed over the entire base of the body, rather than passing 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 lines connecting the support points, the body tips over.
An example of body balance 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 studying 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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To balance a body under the influence arbitrary system forces and pairs of forces, necessary and sufficient to main vector And main point of this system relative to any point were equal to zero. Main vector called geometric sum all forces of the system, and main point relative to a point - the geometric sum of the moments of all forces relative to this point.
In general, the equilibrium conditions in vector form have the form:
Projecting vector equalities (12.1) onto the coordinate axes, we obtain analytical equilibrium conditions:
;
Thus, for the equilibrium of an arbitrary spatial system of forces, it is necessary and sufficient that the sum of the projections of all forces onto each of the three coordinate axes and the sum of their moments relative to each of these axes are equal to zero.
When considering particular cases when the system of forces acting on a body is not arbitrary spatial, the equilibrium conditions are written taking into account the specifics of this system of forces.
Statics problems on body balance under action various systems forces should be solved in the proposed sequence:
1) choose an object of equilibrium;
2) depict everything active forces, acting on the object of equilibrium;
3) discard the connections imposed on the object of equilibrium and replace their action with reactions corresponding to the types of connections;
4) write down a system of equilibrium equations for the resulting system of forces, solve this system and determine the required quantities.
Notes:
■ a material point, a body or a set of interconnected bodies can be chosen as an object (objects) of equilibrium in such a way that all the required forces or part of them are applied to this object (objects);
■ if it is impossible to unambiguously determine all the required forces or other forces from the equilibrium equation unknown parameters, then the task is statically indeterminate and it cannot be solved within the framework of statics. In this case, the following cases are possible: number of unknowns more number equations of statics, the matrix of a system of equations when the number of unknowns is equal to the number of equations is special ( degenerate), number of unknowns less number equations. IN the latter case an object can be in equilibrium only under conditions imposed by active forces.
1.4. Center of parallel forces. Center of gravity
In statics they prove that if a system of parallel forces has a resultant force, then there is a point, and only one, through which its line of action passes. This point is called center of parallel forces . The center of parallel forces has one important property - if all forces are rotated relative to parallel axes passing through the points of their application by the same angle, then the resultant system of these forces will rotate by the same angle relative to a similar axis passing through the center of parallel forces.
Let us consider a body of arbitrary shape located in the field of gravity of the Earth. In this case, each elementary volume of the body under consideration is affected by the force of gravity
,
(1.3)
Where 
– specific gravity volume element 
,
.
When the body is homogeneous, 
does not depend on coordinates.
The forces of gravity acting on each elementary volume of the body are directed towards the center of the Earth. If the size of the body in relation to the size of the Earth is neglected, then the system of gravity forces can be considered a system of parallel forces directed in one direction. Such a system always has a resultant, and, consequently, a center of parallel forces.
The center of the system of gravity forces acting on a body from the Earth is called center of gravity of the body . If a body is considered in a reference system centered at the point ABOUT and with coordinate axes x,y,z(Fig. 1.8), then the radius vector of the center of gravity and its coordinates are determined by the formula:
Here 
– modulus of gravity acting on an elementary volume 
.
The center of gravity does not change its position relative to the body at any orientation relative to the Earth. The center of gravity is a geometric point that may not belong to the body, but is necessarily rigidly connected to it. If the body is homogeneous, i.e. 
, Where 
, then instead of the concept of center of gravity, we can use the center of gravity of the volume occupied by the body. Similarly, if a homogeneous body is a thin plate or shell of constant thickness, or a thin curved rod of constant thickness, then the center of gravity of such a body is called center of gravity of the surface
or lines
.
The formulas by which the coordinates of the centers of gravity of homogeneous bodies are determined are as follows:
– center of gravity of the volume
– center of gravity of the surface
– center of gravity of the line
,
(1.7)
where respectively the values are: V– volume of bodies; S– body surface area; L– body lengths over which integrals are taken.
To find the centers of gravity of bodies, the directly given formulas are used, as well as symmetry rules and partitioning methods complex bodies into simpler ones, for which it is easier to determine the positions of their centers of gravity. In some cases, the positions of the centers of gravity of bodies are found experimentally.
1.5 .Dry friction. Coulomb's Laws
The concepts of dry friction are introduced into theoretical mechanics from physics. Real bodies are not perfectly smooth and completely solid. Therefore, when trying to move or roll one body along the surface of another, in addition to interaction forces directed along the common normal to the contacting surfaces at the point of their contact, forces and pairs of forces arise that prevent sliding and rolling. These forces are called accordingly sliding friction forces and rolling friction forces. Friction is called dry , if there is no lubricant between the interacting solids.
Many statics problems cannot be solved without taking into account friction forces. For example, without these forces, equilibrium of a rigid body on an inclined plane is impossible. Everyone knows the fact that car wheels slip on a slippery road, so the movement itself in most cases is caused by friction forces. Sliding friction and rolling friction are taken into account statically using empirical (experimental) data, which are called Coulomb's laws .
When one body tries to roll on the surface of another, rolling resistance is exerted by a pair of forces called moment of rolling friction forces . Let us formulate Coulomb's laws for rolling friction. The direction of the moment of rolling friction forces is opposite to the direction in which the active forces tend to roll the body. The rolling friction moment is in the range 0 ≤ M tr ≤ M tr.pr. It is determined by the formula
M tr.pr = δ N,
where δ – rolling friction coefficient , having the dimension of length; N– normal pressure. It has been experimentally established that the value of δ depends on the materials of the bodies and the radius of the rolling body. Values for δ can be found in reference books.
A distinctive feature of statics problems in the presence of friction forces is that when the friction force F tr or moment of frictional forces M tr is less than the limit values, the reaction of the bonds, including the force and moment of the friction forces, is determined from the equilibrium equations, as usual. If the friction forces reach limiting values, then they are found using friction coefficients and entered as known quantities. In this case, however, the body is not in equilibrium and the application of static equations to the entire body becomes unlawful. To establish the equilibrium of bodies in the presence of friction, the equilibrium equations are supplemented with the corresponding inequalities, which require that the sliding friction force or the rolling friction moment do not exceed the limit values.
Questions for self-control
1. What is studied in the statics section of the theoretical mechanics course?
2. What is called an absolutely rigid body?
3. How are the concepts of force and systems of forces defined in statics?
4. What relationships exist between forces and systems of forces? Give a classification of forces.
5. What axioms are the theoretical principles of statics based on?
6. Which body is called unfree?
7. How are the concepts of connections and their reactions defined?
8. What basic connections can be imposed on an absolutely rigid body? What reactions occur in these connections?
9. How are the equilibrium conditions of an absolutely rigid body formulated in vector and analytical forms?
10. What is the sequence of solving the problem of determining the reactions of bonds?
11. What conditions must be met for the system of equilibrium equations of an absolutely rigid body to be solvable?
12. How are the radius vector and coordinates of the center of gravity of a body determined?
13. How do statics take into account the action of dry friction forces on a solid body?
14. What are the features of solving statics problems in the presence of friction forces?
Definition
The equilibrium of a body is a state when any acceleration of the body is equal to zero, that is, all the actions of forces and moments of forces on the body are balanced. In this case, the body can:
- be in a state of calm;
- move evenly and straightly;
- rotate uniformly around an axis that passes through its center of gravity.
Body equilibrium conditions
If the body is in equilibrium, then two conditions are simultaneously satisfied.
- The vector sum of all forces acting on the body is equal to the zero vector: $\sum_n((\overrightarrow(F))_n)=\overrightarrow(0)$
- The algebraic sum of all moments of forces acting on the body is equal to zero: $\sum_n(M_n)=0$
Two equilibrium conditions are necessary but not sufficient. Let's give an example. Let us consider a wheel rolling uniformly without slipping on a horizontal surface. Both equilibrium conditions are satisfied, but the body moves.
Let's consider the case when the body does not rotate. In order for the body not to rotate and to be in equilibrium, it is necessary that the sum of the projections of all forces on an arbitrary axis equals zero, that is, the resultant of the forces. Then the body is either at rest or moving evenly and in a straight line.
A body that has an axis of rotation will be in equilibrium if the rule of moments of forces is satisfied: the sum of the moments of forces that rotate the body clockwise must be equal to the sum of the moments of forces that rotate it counterclockwise.
To obtain right moment at with the least effort, you need to apply force as far as possible from the axis of rotation, thereby increasing the leverage of the force and correspondingly decreasing the value of the force. Examples of bodies that have an axis of rotation are: levers, doors, blocks, rotators, and the like.
Three types of equilibrium of bodies that have a fulcrum
- stable equilibrium, if the body, being removed from the equilibrium position to the next closest position and left at rest, returns to this position;
- unstable equilibrium, if the body, being taken from the equilibrium position to an adjacent position and left at rest, will deviate even more from this position;
- indifferent equilibrium - if the body, being brought to an adjacent position and left calm, remains in its new position.
Equilibrium of a body with a fixed axis of rotation
- stable if in the equilibrium position the center of gravity C occupies the lowest position of all possible nearby positions, and its potential energy will have smallest value of all possible values in adjacent positions;
- unstable if the center of gravity C occupies the highest of all nearby positions, and the potential energy has the greatest value;
- indifferent if the center of gravity of the body C in all nearby possible positions is at the same level, and the potential energy does not change during the transition of the body.
Problem 1
Body A with mass m = 8 kg is placed on a rough horizontal table surface. A thread is tied to the body, thrown over block B (Figure 1, a). What weight F can be tied to the end of the thread hanging from the block so as not to upset the balance of body A? Friction coefficient f = 0.4; Neglect friction on the block.
Let us determine the weight of the body ~A: ~G = mg = 8$\cdot $9.81 = 78.5 N.
We assume that all forces are applied to body A. When the body is placed on a horizontal surface, then only two forces act on it: weight G and the oppositely directed reaction of the support RA (Fig. 1, b).
If we apply some force F acting along a horizontal surface, then the reaction RA, balancing the forces G and F, will begin to deviate from the vertical, but body A will be in equilibrium until the modulus of force F exceeds maximum value friction force Rf max corresponding to the limiting value of the angle $(\mathbf \varphi )$o (Fig. 1, c).
By decomposing the reaction RA into two components Rf max and Rn, we obtain a system of four forces applied to one point (Fig. 1, d). By projecting this system of forces onto the x and y axes, we obtain two equilibrium equations:
$(\mathbf \Sigma )Fkx = 0, F - Rf max = 0$;
$(\mathbf \Sigma )Fky = 0, Rn - G = 0$.
We solve the resulting system of equations: F = Rf max, but Rf max = f$\cdot $ Rn, and Rn = G, so F = f$\cdot $ G = 0.4$\cdot $ 78.5 = 31.4 N; m = F/g = 31.4/9.81 = 3.2 kg.
Answer: Cargo mass t = 3.2 kg
Problem 2
The system of bodies shown in Fig. 2 is in a state of equilibrium. Cargo weight tg=6 kg. The angle between the vectors is $\widehat((\overrightarrow(F))_1(\overrightarrow(F))_2)=60()^\circ $. $\left|(\overrightarrow(F))_1\right|=\left|(\overrightarrow(F))_2\right|=F$. Find the mass of the weights.
The resultant forces $(\overrightarrow(F))_1and\ (\overrightarrow(F))_2$ are equal in magnitude to the weight of the load and opposite to it in direction: $\overrightarrow(R)=(\overrightarrow(F))_1+(\overrightarrow (F))_2=\ -m\overrightarrow(g)$. By the cosine theorem, $(\left|\overrightarrow(R)\right|)^2=(\left|(\overrightarrow(F))_1\right|)^2+(\left|(\overrightarrow(F) )_2\right|)^2+2\left|(\overrightarrow(F))_1\right|\left|(\overrightarrow(F))_2\right|(cos \widehat((\overrightarrow(F)) _1(\overrightarrow(F))_2)\ )$.
Hence $(\left(mg\right))^2=$; $F=\frac(mg)(\sqrt(2\left(1+(cos 60()^\circ \ )\right)))$;
Since the blocks are movable, then $m_g=\frac(2F)(g)=\frac(2m)(\sqrt(2\left(1+\frac(1)(2)\right)))=\frac(2 \cdot 6)(\sqrt(3))=6.93\ kg\ $
Answer: the mass of each weight is 6.93 kg
