Statics is a branch of mechanics that studies the equilibrium of bodies. Statics makes it possible to determine the conditions of equilibrium of bodies and answers some questions that relate to the movement of bodies, for example, it 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 forces. Different amounts of forces act on bodies, but if we find the resultant force, for a body in equilibrium it will be equal to zero.
There are:
- static equilibrium - the body is at rest;
- dynamic equilibrium - a body moves at a constant speed.
Static balance. If forces F1, F2, F3, and so on act on a body, then the main requirement for the existence of a state of equilibrium is (equilibrium). This is a vector equation in three-dimensional space, and represents three separate equations, one for each direction of 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 motion at the same time: a barrel rolling down an inclined plane, a dancing couple. With such a movement, the condition of equilibrium alone is not enough.
The necessary equilibrium condition in this case will be:
In practice and in life, the stability of bodies, which characterizes balance, plays an important role.
There are different types of balance:
- Stable balance;
- Unstable equilibrium;
- Indifferent balance.
Stable equilibrium is an 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 Vanka-Vstanka or Tumbler, laundry on a line are in a state of stable equilibrium).
Unstable equilibrium is a state when a body, after being removed from an equilibrium position, deviates due to the resulting force even further from the equilibrium position (a tennis ball on a convex surface).
Indifferent equilibrium - when 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 hanging on a nail are in a state of indifferent equilibrium). The axis of rotation and the center of gravity coincide.
For two bodies, the body will be more stable, which has a larger support area.
Equilibrium of a mechanical system- this is a state in which all points of a mechanical system are at rest with respect to the reference system under consideration. If the reference frame is inertial, equilibrium is called absolute, if non-inertial - relative.
To find the equilibrium conditions of an absolutely rigid body, it is necessary to mentally break it down into a large number of fairly 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 on 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 a 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 a body to be in equilibrium, it is necessary and sufficient that the geometric sum of all forces acting on any element of this body be equal to zero.
From this 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 up the equation for all elements of the body:
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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 magnitude and opposite in direction.
Hence,
.
The first condition for the equilibrium of a rigid body(systems of bodies) is the equality to zero of the geometric sum of all external forces applied to the body.
This condition is necessary, but not sufficient. This is easy to verify by remembering the rotating action of a pair of forces, the geometric sum of which is also 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 of a rigid body in the case of an arbitrary number of external forces look like this:
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Physics, 10th grade
Lesson 14. Statics. Equilibrium of absolutely rigid bodies
List of questions covered in the lesson:
1. Conditions for body balance
2.Moment of force
3.Shoulder strength
4. Center of gravity
Glossary on the topic
Statics– the branch of mechanics in which the equilibrium of absolutely rigid bodies is studied is called statics
Absolutely rigid body– a model concept of classical mechanics, denoting a set of points whose distances between their current positions do not change.
Center of gravity– the center of gravity of a body is the point through which, at any position of the body in space, the resultant of the forces of gravity acting on all particles of the body passes.
Shoulder of power
Moment of power - this is a physical quantity equal to the product of the force modulus and its arm.
Stable balance- this is an equilibrium in which a body, removed from a state of stable equilibrium, tends to return to its initial position.
Unstable equilibrium- this is an equilibrium in which a body, taken out of an equilibrium position and left to itself, will deviate even more from the equilibrium position.
Indifferent equilibrium of the system- equilibrium in which, after eliminating the causes that caused small deviations, the system remains at rest in this rejected state
Basic and additional literature on the topic of the lesson:
Myakishev G.Ya., Bukhovtsev B.B., Sotsky N.N. Physics. 10th grade. Textbook for general education organizations M.: Prosveshchenie, 2017. – P. 165 – 169.
Rymkevich A.P. Collection of problems in physics. 10-11 grade. - M.: Bustard, 2009.
Stepanova G.N. Collection of problems in physics. 10-11 grade. - M.: Enlightenment. 1999, pp. 48-50.
Theoretical material for self-study
Equilibrium is a state of rest, i.e. if a body is at rest relative to an inertial frame of reference, then it is said to be in equilibrium. Questions of balance are of interest to builders, climbers, circus performers and many, many other people. Every person has had to deal with the problem of maintaining balance. Why do some bodies, when disturbed from a state of equilibrium, fall, while others do not? Let us find out under what conditions the body will be in a state of equilibrium.
The branch of mechanics in which the equilibrium of absolutely rigid bodies is studied is called statics. Statics is a special case of dynamics. In statics, a solid body is considered as absolutely solid, i.e. non-deformable body. This means that the deformation is so small that it can be ignored.
A center of gravity exists for any body. This point can also be located outside the body. How to hang or support the body so that it is in balance.
Archimedes solved a similar problem in his time. He also introduced the concept of leverage and moment of force.
Shoulder of power- this is the length of the perpendicular lowered from the axis of rotation to the line of action of the force.
Moment of power is a physical quantity equal to the product of the force modulus and its shoulder.
After his research, Archimedes formulated the condition for the equilibrium of a lever and derived the formula:
This rule is a consequence of Newton's 2nd law.
First equilibrium condition
For a body to balance, it is necessary that the sum of all forces applied to the body be equal to zero.
the formula must be in vector form and have a sum sign
Second equilibrium condition
When a rigid body is in equilibrium, the sum of the moments of all external forces acting on it relative to any axis is equal to zero.
No less important is the case when the body has a support area. A body having a support area is in equilibrium when the vertical line passing through the center of gravity of the body does not extend beyond the support area of this body. It is known that there is a leaning tower in the city of Pisa in Italy. Even though the tower is tilted, it does not topple, although it is often called leaning. It is obvious that with the inclination that the tower has achieved so far, the vertical drawn from the center of gravity of the tower still runs inside its support area.
In practice, an important role is played not only by the fulfillment of the condition of equilibrium of bodies, but also by the qualitative characteristic of equilibrium, called stability.
There are 3 types of equilibrium: stable, unstable, indifferent.
If, when a body deviates from an equilibrium position, forces or moments of force arise that tend to return the body to an equilibrium position, then such equilibrium is called stable.
Unstable equilibrium is the opposite case. When a body deviates from its equilibrium position, forces or moments of force arise that tend to increase this deviation.
Finally, if, even with a small deviation from the equilibrium position, the body still remains in equilibrium, then such equilibrium is called indifferent.
Most often it is necessary for the balance to be stable. When the balance is disturbed, the structure becomes dangerous if its size is large.
Examples and analysis of problem solving
1 . What is the moment of gravity of a load weighing 40 kg suspended on the bracket ABC, relative to the axis passing through point B, if AB = 0.5 m and angle α = 45 0
The moment of force is a value equal to the product of the force modulus and its arm.
First, let's find the arm of the force; to do this, we need to lower the perpendicular from the fulcrum to the line of action of the force. The gravity arm is equal to the distance AC. Since the angle is 45°, we see that AC = AB
We find the gravity module using the formula:
After substituting the numerical values of the quantities, we get:
F=40×9.8 =400 N, M= 400 ×0.5=200 N m.
Answer: M=200 N m.
2 . By applying a vertical force F, a load of mass M - 100 kg is held in place using a lever (see figure). The lever consists of a frictionless hinge and a homogeneous massive rod with a length of L = 8 m. The distance from the hinge axis to the point of suspension of the load is b = 2 m. What is the force module F equal if the mass of the lever is 40 kg.
According to the conditions of the problem, the lever is in equilibrium. Let us write the second equilibrium condition for the lever:
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After substituting the numerical values of the quantities, we get
F= (100×9.8 ×2 + 0.5×40×9.8×8)/8=450 N
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 body of a certain 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. The unit of measurement of moment of force in the international SI system is 1 Newtonmeter.
Definition. Rule of Moments
If the algebraic sum of all moments applied to a body relative to a fixed axis of rotation is equal to 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 are different types of equilibrium. Thus, a distinction is made 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. In 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 the 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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DEFINITION
Stable balance- this is an equilibrium in which a body, removed from a position of equilibrium and left to itself, returns to its previous position.
This occurs if, with a slight displacement of the body in any direction from the original position, the resultant of the forces acting on the body becomes non-zero and is directed towards the equilibrium position. For example, a ball lying at the bottom of a spherical depression (Fig. 1 a).
DEFINITION
Unstable equilibrium- this is an equilibrium in which a body, taken out of an equilibrium position and left to itself, will deviate even more from the equilibrium position.
In this case, with a slight displacement of the body from the equilibrium position, the resultant of the forces applied to it is non-zero and directed from the equilibrium position. An example is a ball located at the top point of a convex spherical surface (Fig. 1 b).
DEFINITION
Indifferent Equilibrium- this is an equilibrium in which a body, taken out of an equilibrium position and left to its own devices, does not change its position (state).
In this case, with small displacements of the body from the original position, the resultant of the forces applied to the body remains equal to zero. For example, a ball lying on a flat surface (Fig. 1c).
Fig.1. Different types of body balance on a support: a) stable balance; b) unstable equilibrium; c) indifferent equilibrium.
Static and dynamic balance of bodies
If, as a result of the action of forces, the body does not receive acceleration, it can be at rest or move uniformly in a straight line. Therefore, we can talk about static and dynamic equilibrium.
DEFINITION
Static balance- this is an equilibrium when, under the influence of applied forces, the body is at rest.
Dynamic balance- this is an equilibrium when, due to the action of forces, the body does not change its movement.
A lantern suspended on cables, or any building structure, is in a state of static equilibrium. As an example of dynamic equilibrium, consider a wheel that rolls on a flat surface in the absence of friction forces.
