In this topic we will look at a very special type of irregular motion. Based on the opposition to uniform movement, uneven movement is movement at unequal speed along any trajectory. What is the peculiarity of uniformly accelerated motion? This is an uneven movement, but which "equally accelerated". We associate acceleration with increasing speed. Let's remember the word "equal", we get an equal increase in speed. How do we understand “equal increase in speed”, how can we evaluate whether the speed is increasing equally or not? To do this, we need to record time and estimate the speed over the same time interval. For example, a car starts to move, in the first two seconds it develops a speed of up to 10 m/s, in the next two seconds it reaches 20 m/s, and after another two seconds it already moves at a speed of 30 m/s. Every two seconds the speed increases and each time by 10 m/s. This is uniformly accelerated motion.

The physical quantity that characterizes how much the speed increases each time is called acceleration.

Can the movement of a cyclist be considered uniformly accelerated if, after stopping, in the first minute his speed is 7 km/h, in the second - 9 km/h, in the third - 12 km/h? It is forbidden! The cyclist accelerates, but not equally, first he accelerated by 7 km/h (7-0), then by 2 km/h (9-7), then by 3 km/h (12-9).

Typically, movement with increasing speed is called accelerated movement. Movement with decreasing speed is slow motion. But physicists call any movement with changing speed accelerated movement. Whether the car starts moving (the speed increases!) or brakes (the speed decreases!), in any case it moves with acceleration.

Uniformly accelerated motion- this is the movement of a body in which its speed for any equal intervals of time changes(can increase or decrease) the same

Body acceleration

Acceleration characterizes the rate at which speed changes. This is the number by which the speed changes every second. If the acceleration of a body is large in magnitude, this means that the body quickly gains speed (when it accelerates) or quickly loses it (when braking). Acceleration is a physical vector quantity, numerically equal to the ratio of the change in speed to the period of time during which this change occurred.

Let's determine the acceleration in the next problem. At the initial moment of time, the speed of the ship was 3 m/s, at the end of the first second the speed of the ship became 5 m/s, at the end of the second - 7 m/s, at the end of the third 9 m/s, etc. Obviously, . But how did we determine? We are looking at the speed difference over one second. In the first second 5-3=2, in the second second 7-5=2, in the third 9-7=2. But what if the speeds are not given for every second? Such a problem: the initial speed of the ship is 3 m/s, at the end of the second second - 7 m/s, at the end of the fourth 11 m/s. In this case, you need 11-7 = 4, then 4/2 = 2. We divide the speed difference by the time period.

This formula is most often used in a modified form when solving problems:

The formula is not written in vector form, so we write the “+” sign when the body is accelerating, the “-” sign when it is slowing down.

Acceleration vector direction

The direction of the acceleration vector is shown in the figures

In this figure, the car moves in a positive direction along the Ox axis, the velocity vector always coincides with the direction of movement (directed to the right). When the acceleration vector coincides with the direction of the speed, this means that the car is accelerating. Acceleration is positive.

During acceleration, the direction of acceleration coincides with the direction of speed. Acceleration is positive.

In this picture, the car is moving in the positive direction along the Ox axis, the velocity vector coincides with the direction of movement (directed to the right), the acceleration does NOT coincide with the direction of the speed, this means that the car is braking. Acceleration is negative.

When braking, the direction of acceleration is opposite to the direction of speed. Acceleration is negative.

Let's figure out why the acceleration is negative when braking. For example, in the first second the ship slowed down from 9m/s to 7m/s, in the second second to 5m/s, in the third to 3m/s. The speed changes to "-2m/s". 3-5=-2; 5-7=-2; 7-9=-2m/s. This is where the negative acceleration value comes from.

When solving problems, if the body slows down, acceleration is substituted into the formulas with a minus sign!!!

Moving during uniformly accelerated motion

An additional formula called timeless

Formula in coordinates

Medium speed communication

With uniformly accelerated motion, the average speed can be calculated as the arithmetic mean of the initial and final speeds

From this rule follows a formula that is very convenient to use when solving many problems

Path relationship

If a body moves uniformly accelerated, the initial speed is zero, then the paths traversed in successive equal intervals of time are related as a successive series of odd numbers.

The main thing to remember

1) What is uniformly accelerated motion;
2) What characterizes acceleration;
3) Acceleration is a vector. If a body accelerates, the acceleration is positive, if it slows down, the acceleration is negative;
3) Direction of the acceleration vector;
4) Formulas, units of measurement in SI

Exercises

Two trains are moving towards each other: one is heading north at an accelerated rate, the other is moving slowly to the south. How are train accelerations directed?

Equally to the north. Because the first train's acceleration coincides in direction with the movement, and the second train's acceleration is opposite to the movement (it slows down).

Topics of the Unified State Examination codifier: types of mechanical motion, speed, acceleration, equations of rectilinear uniformly accelerated motion, free fall.

Uniformly accelerated motion - this is movement with a constant acceleration vector. Thus, with uniformly accelerated motion, the direction and absolute magnitude of the acceleration remain unchanged.

Dependence of speed on time.

When studying uniform rectilinear motion, the question of the dependence of speed on time did not arise: the speed was constant during the movement. However, with uniformly accelerated motion, the speed changes over time, and we have to find out this dependence.

Let's practice some basic integration again. We proceed from the fact that the derivative of the velocity vector is the acceleration vector:

. (1)

In our case we have . What needs to be differentiated to get a constant vector? Of course, the function. But not only that: you can add an arbitrary constant vector to it (after all, the derivative of a constant vector is zero). Thus,

. (2)

What is the meaning of the constant? At the initial moment of time, the speed is equal to its initial value: . Therefore, assuming in formula (2) we get:

So, the constant is the initial speed of the body. Now relation (2) takes its final form:

. (3)

In specific problems, we choose a coordinate system and move on to projections onto coordinate axes. Often two axes and a rectangular Cartesian coordinate system are enough, and vector formula (3) gives two scalar equalities:

, (4)

. (5)

The formula for the third velocity component, if needed, is similar.)

Law of motion.

Now we can find the law of motion, that is, the dependence of the radius vector on time. We recall that the derivative of the radius vector is the speed of the body:

We substitute here the expression for speed given by formula (3):

(6)

Now we have to integrate equality (6). It is not difficult. To get , you need to differentiate the function. To obtain, you need to differentiate. Let's not forget to add an arbitrary constant:

It is clear that is the initial value of the radius vector at time . As a result, we obtain the desired law of uniformly accelerated motion:

. (7)

Moving on to projections onto coordinate axes, instead of one vector equality (7), we obtain three scalar equalities:

. (8)

. (9)

. (10)

Formulas (8) - (10) give the dependence of the coordinates of the body on time and therefore serve as a solution to the main problem of mechanics for uniformly accelerated motion.

Let's return again to the law of motion (7). Note that - movement of the body. Then
we get the dependence of displacement on time:

Rectilinear uniformly accelerated motion.

If uniformly accelerated motion is rectilinear, then it is convenient to choose a coordinate axis along the straight line along which the body moves. Let, for example, this be the axis. Then to solve the problems we will only need three formulas:

where is the projection of displacement onto the axis.

But very often another formula that is a consequence of them helps. Let us express time from the first formula:

and substitute it into the formula for moving:

After algebraic transformations (be sure to do them!) we arrive at the relation:

This formula does not contain time and allows you to quickly come to an answer in those problems where time does not appear.

Free fall.

An important special case of uniformly accelerated motion is free fall. This is the name given to the movement of a body near the surface of the Earth without taking into account air resistance.

The free fall of a body, regardless of its mass, occurs with a constant free fall acceleration directed vertically downward. In almost all problems, m/s is assumed in calculations.

Let's look at several problems and see how the formulas we derived for uniformly accelerated motion work.

Task. Find the landing speed of a raindrop if the height of the cloud is km.

Solution. Let's direct the axis vertically downwards, placing the origin at the point of separation of the drop. Let's use the formula

We have: - the required landing speed, . We get: , from . We calculate: m/s. This is 720 km/h, about the speed of a bullet.

In fact, raindrops fall at speeds of the order of several meters per second. Why is there such a discrepancy? Windage!

Task. A body is thrown vertically upward with a speed of m/s. Find its speed in c.

Here, so. We calculate: m/s. This means the speed will be 20 m/s. The projection sign indicates that the body will fly down.

Task. From a balcony located at a height of m, a stone was thrown vertically upward at a speed of m/s. How long will it take for the stone to fall to the ground?

Solution. Let's direct the axis vertically upward, placing the origin on the surface of the Earth. We use the formula

We have: so , or . Solving the quadratic equation, we get c.

Horizontal throw.

Uniformly accelerated motion is not necessarily linear. Consider the motion of a body thrown horizontally.

Suppose that a body is thrown horizontally with a speed from a height. Let's find the time and flight range, and also find out what trajectory the movement takes.

Let us choose a coordinate system as shown in Fig.

1 .

We use the formulas:

. (11)

In our case . We get:

We find the flight time from the condition that at the moment of fall the coordinate of the body becomes zero:

Flight range is the coordinate value at the moment of time:

We obtain the trajectory equation by excluding time from equations (11). We express from the first equation and substitute it into the second:

We obtained a dependence on , which is the equation of a parabola. Consequently, the body flies in a parabola.

Throw at an angle to the horizontal.

Let's consider a slightly more complex case of uniformly accelerated motion: the flight of a body thrown at an angle to the horizon.

Let us assume that a body is thrown from the surface of the Earth with a speed directed at an angle to the horizon. Let's find the time and flight range, and also find out what trajectory the body is moving along.

Let us choose a coordinate system as shown in Fig.

2.

We start with the equations: (Be sure to do these calculations yourself!) As you can see, the dependence on is again a parabolic equation. Try also to show that the maximum lift height is given by the formula. In general uniformly accelerated motion called such a movement in which the acceleration vector remains unchanged in magnitude and direction. An example of such movement is the movement of a stone thrown at a certain angle to the horizon (without taking into account air resistance). At any point in the trajectory, the acceleration of the stone is equal to the acceleration of gravity. For a kinematic description of the movement of a stone, it is convenient to choose a coordinate system so that one of the axes, for example the axis OY, was directed parallel to the acceleration vector. Then the curvilinear movement of the stone can be represented as the sum of two movements - uniformly accelerated motion rectilinear uniformly accelerated motion uniform rectilinear motion in the perpendicular direction, i.e. along the axis OX(Fig. 1.4.1).

Thus, the study of uniformly accelerated motion is reduced to the study of rectilinear uniformly accelerated motion. In the case of rectilinear motion, the velocity and acceleration vectors are directed along the straight line of motion. Therefore, the speed υ and acceleration a in projections onto the direction of movement can be considered as algebraic quantities.

Figure 1.4.1. Projections of velocity and acceleration vectors onto coordinate axes. ax = 0, ay = -g

In uniformly accelerated rectilinear motion, the speed of a body is determined by the formula

(*)

In this formula, υ 0 is the speed of the body at t = 0 (starting speed ), a= const - acceleration. On the speed graph υ ( t) this dependence looks like a straight line (Fig. 1.4.2).

Figure 1.4.2. Speed ​​graphs of uniformly accelerated motion

Acceleration can be determined from the slope of the velocity graph a bodies. The corresponding constructions are shown in Fig. 1.4.2 for graph I. Acceleration is numerically equal to the ratio of the sides of the triangle ABC:

The greater the angle β that the velocity graph forms with the time axis, i.e., the greater the slope of the graph ( steepness), the greater the acceleration of the body.

For graph I: υ 0 = -2 m/s, a= 1/2 m/s 2.

For schedule II: υ 0 = 3 m/s, a= -1/3 m/s 2

The velocity graph also allows you to determine the projection of movement s bodies for some time t. Let us select on the time axis a certain small period of time Δ t. If this period of time is small enough, then the change in speed over this period is small, i.e. the movement during this period of time can be considered uniform with a certain average speed, which is equal to the instantaneous speed υ of the body in the middle of the interval Δ t. Therefore, the displacement Δ s in time Δ t will be equal to Δ s = υΔ t. This movement is equal to the area of ​​the shaded strip (Fig. 1.4.2). Breaking down the time period from 0 to some point t for small intervals Δ t, we find that the movement s for a given time t with uniformly accelerated rectilinear motion is equal to the area of ​​the trapezoid ODEF. The corresponding constructions were made for graph II in Fig. 1.4.2. Time t taken equal to 5.5 s.

Since υ - υ 0 = at, the final formula for moving s body with uniformly accelerated motion over a time interval from 0 to t will be written in the form:

(**)

To find the coordinates y body at any time t need to the starting coordinate y 0 add movement in time t:

(***)

This expression is called law of uniformly accelerated motion .

When analyzing uniformly accelerated motion, sometimes the problem arises of determining the movement of a body based on the given values ​​of the initial υ 0 and final υ velocities and acceleration a. This problem can be solved using the equations written above by eliminating time from them t. The result is written in the form

From this formula we can obtain an expression for determining the final speed υ of a body if the initial speed υ 0 and acceleration are known a and moving s:

If the initial speed υ 0 is zero, these formulas take the form

It should be noted once again that the quantities υ 0, υ, included in the formulas for uniformly accelerated rectilinear motion s, a, y 0 are algebraic quantities. Depending on the specific type of movement, each of these quantities can take on both positive and negative values.

In previous lessons, we discussed how to determine the distance traveled during uniform linear motion. It's time to find out how to determine the coordinates of a body, the distance traveled and displacement during rectilinear uniformly accelerated motion. This can be done if we consider rectilinear uniformly accelerated motion as a set of a large number of very small uniform displacements of the body.

The first to solve the problem of the location of a body at a certain moment in time during accelerated motion was the Italian scientist Galileo Galilei (Fig. 1).

Rice. 1. Galileo Galilei (1564-1642)

He conducted his experiments with an inclined plane. He launched a ball, a musket bullet, along the chute, and then determined the acceleration of this body. How did he do it? He knew the length of the inclined plane, and determined the time by the beat of his heart or pulse (Fig. 2).

Rice. 2. Galileo's experiment

Consider the speed dependence graph uniformly accelerated linear motion from time. You know this dependence; it is a straight line: .

Rice. 3. Determination of displacement during uniformly accelerated linear motion

We divide the speed graph into small rectangular sections (Fig. 3). Each section will correspond to a certain speed, which can be considered constant in a given period of time. It is necessary to determine the distance traveled during the first period of time. Let's write the formula: . Now let's calculate the total area of ​​all the figures we have.

The sum of the areas during uniform motion is the total distance traveled.

Please note: the speed will change from point to point, thereby we will get the path traveled by the body precisely during rectilinear uniformly accelerated motion.

Note that during rectilinear uniformly accelerated motion of a body, when speed and acceleration are directed in the same direction (Fig. 4), the displacement module is equal to the distance traveled, therefore, when we determine the displacement module, we determine distance traveled. In this case, we can say that the displacement module will be equal to the area of ​​the figure, limited by the graph of speed and time.

Rice. 4. The displacement module is equal to the distance traveled

Let's use mathematical formulas to calculate the area of ​​the indicated figure.

Rice. 5 Illustration for calculating area

The area of ​​the figure (numerically equal to the distance traveled) is equal to half the sum of the bases multiplied by the height. Please note that in the figure, one of the bases is the initial speed, and the second base of the trapezoid will be the final speed, indicated by the letter . The height of the trapezoid is equal to , this is the period of time during which the movement occurred.

We can write the final velocity, discussed in the previous lesson, as the sum of the initial velocity and the contribution due to the constant acceleration of the body. The resulting expression is:

If you open the brackets, it becomes double. We can write the following expression:

If you write each of these expressions separately, the result will be the following:

This equation was first obtained through the experiments of Galileo Galilei. Therefore, we can consider that it was this scientist who first made it possible to determine the location of a body during rectilinear uniformly accelerated motion at any time. This is the solution to the main problem of mechanics.

Now let's remember that the distance traveled, equal in our case movement module, is expressed by the difference:

If we substitute this expression into Galileo’s equation, we obtain a law according to which the coordinate of a body changes during rectilinear uniformly accelerated motion:

It should be remembered that the quantities are projections of velocity and acceleration onto the selected axis. Therefore, they can be both positive and negative.

Conclusion

The next stage of consideration of movement will be the study of movement along a curvilinear trajectory.

Bibliography

1. Kikoin I.K., Kikoin A.K. Physics: textbook for 9th grade of high school. - M.: Enlightenment.
2. Peryshkin A.V., Gutnik E.M., Physics. 9th grade: textbook for general education. institutions/A. V. Peryshkin, E. M. Gutnik. - 14th ed., stereotype. - M.: Bustard, 2009. - 300.
3. Sokolovich Yu.A., Bogdanova G.S.. Physics: A reference book with examples of problem solving. - 2nd edition repartition. - X.: Vesta: Ranok Publishing House, 2005. - 464 p.

Additional recommended links to Internet resources

1. Internet portal “class-fizika.narod.ru” ()
2. Internet portal “videouroki.net” ()
3. Internet portal “foxford.ru” ()

Homework

1. Write down the formula that determines the projection of the displacement vector of a body during rectilinear uniformly accelerated motion.
2. A cyclist, whose initial speed is 15 km/h, slides down a hill in 5 s. Determine the length of the slide if the cyclist moved with a constant acceleration of 0.5 m/s^2 .
3. How do the dependences of displacement on time differ for uniform and uniformly accelerated motion?

The part of mechanics in which motion is studied without considering the reasons causing this or that character of motion is called kinematics.
Mechanical movement called a change in the position of a body relative to other bodies
Reference system called the body of reference, the coordinate system associated with it and the clock.
Body of reference name the body relative to which the position of other bodies is considered.
Material point is a body whose dimensions can be neglected in this problem.
Trajectory called a mental line that a material point describes during its movement.

According to the shape of the trajectory, the movement is divided into:
A) rectilinear- the trajectory is a straight line segment;
b) curvilinear- the trajectory is a segment of a curve.

Path is the length of the trajectory that a material point describes over a given period of time. This is a scalar quantity.
Moving is a vector connecting the initial position of a material point with its final position (see figure).

It is very important to understand how a path differs from a movement. The most important difference is that movement is a vector with a beginning at the point of departure and an end at the destination (it does not matter at all what route this movement took). And the path is, on the contrary, a scalar quantity that reflects the length of the trajectory traveled.

Uniform linear movement called a movement in which a material point makes the same movements over any equal periods of time
Speed ​​of uniform linear motion is called the ratio of movement to the time during which this movement occurred:

For uneven motion they use the concept average speed. Average speed is often introduced as a scalar quantity. This is the speed of such uniform motion in which the body travels the same path in the same time as during uneven motion:

Instant speed call the speed of a body at a given point in the trajectory or at a given moment in time.
Uniformly accelerated linear motion- this is a rectilinear movement in which the instantaneous speed for any equal periods of time changes by the same amount

Acceleration is the ratio of the change in the instantaneous speed of a body to the time during which this change occurred:

The dependence of the body coordinates on time in uniform rectilinear motion has the form: x = x 0 + V x t, where x 0 is the initial coordinate of the body, V x is the speed of movement.
Free fall called uniformly accelerated motion with constant acceleration g = 9.8 m/s 2, independent of the mass of the falling body. It occurs only under the influence of gravity.

Free fall speed is calculated using the formula:

Vertical movement is calculated using the formula:

One type of motion of a material point is motion in a circle. With such movement, the speed of the body is directed along a tangent drawn to the circle at the point where the body is located (linear speed). You can describe the position of a body on a circle using a radius drawn from the center of the circle to the body. The displacement of a body when moving in a circle is described by rotating the radius of the circle connecting the center of the circle with the body. The ratio of the angle of rotation of the radius to the period of time during which this rotation occurred characterizes the speed of movement of the body in a circle and is called angular velocity ω:

Angular velocity is related to linear velocity by the relation

where r is the radius of the circle.
The time it takes a body to complete a complete revolution is called circulation period. The reciprocal of the period is the circulation frequency - ν

Since during uniform motion in a circle the velocity module does not change, but the direction of the velocity changes, with such motion there is acceleration. He is called centripetal acceleration, it is directed radially towards the center of the circle:

Basic concepts and laws of dynamics

The part of mechanics that studies the reasons that caused the acceleration of bodies is called dynamics

Newton's first law:
There are reference systems relative to which a body maintains its speed constant or is at rest if other bodies do not act on it or the action of other bodies is compensated.
The property of a body to maintain a state of rest or uniform linear motion with balanced external forces acting on it is called inertia. The phenomenon of maintaining the speed of a body under balanced external forces is called inertia. Inertial reference systems are systems in which Newton's first law is satisfied.

Galileo's principle of relativity:
in all inertial reference systems under the same initial conditions, all mechanical phenomena proceed in the same way, i.e. subject to the same laws
Weight is a measure of the inertia of a body
Force is a quantitative measure of the interaction of bodies.

Newton's second law:
The force acting on a body is equal to the product of the mass of the body and the acceleration imparted by this force:
$F↖(→) = m⋅a↖(→)$

The addition of forces consists of finding the resultant of several forces, which produces the same effect as several simultaneously acting forces.

Newton's third law:
The forces with which two bodies act on each other are located on the same straight line, equal in magnitude and opposite in direction:
$F_1↖(→) = -F_2↖(→) $

Newton's III law emphasizes that the action of bodies on each other is in the nature of interaction. If body A acts on body B, then body B acts on body A (see figure).

Or in short, the force of action is equal to the force of reaction. The question often arises: why does a horse pull a sled if these bodies interact with equal forces? This is possible only through interaction with the third body - the Earth. The force with which the hooves press into the ground must be greater than the frictional force of the sled on the ground. Otherwise, the hooves will slip and the horse will not move.
If a body is subjected to deformation, forces arise that prevent this deformation. Such forces are called elastic forces.

Hooke's law written in the form

where k is the spring stiffness, x is the deformation of the body. The “−” sign indicates that the force and deformation are directed in different directions.

When bodies move relative to each other, forces arise that impede the movement. These forces are called friction forces. A distinction is made between static friction and sliding friction. Sliding friction force calculated by the formula

where N is the support reaction force, µ is the friction coefficient.
This force does not depend on the area of ​​the rubbing bodies. The friction coefficient depends on the material from which the bodies are made and the quality of their surface treatment.

Static friction occurs if the bodies do not move relative to each other. The static friction force can vary from zero to a certain maximum value

By gravitational forces are the forces with which any two bodies are attracted to each other.

Law of universal gravitation:
any two bodies are attracted to each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

Here R is the distance between the bodies. The law of universal gravitation in this form is valid either for material points or for spherical bodies.

Body weight called the force with which the body presses on a horizontal support or stretches the suspension.

Gravity- this is the force with which all bodies are attracted to the Earth:

With a stationary support, the weight of the body is equal in magnitude to the force of gravity:

If a body moves vertically with acceleration, then its weight will change.
When a body moves with upward acceleration, its weight

It can be seen that the weight of the body is greater than the weight of the body at rest.

When a body moves with downward acceleration, its weight

In this case, the weight of the body is less than the weight of the body at rest.

Weightlessness is the movement of a body in which its acceleration is equal to the acceleration of gravity, i.e. a = g. This is possible if only one force acts on the body - gravity.
Artificial Earth satellite- this is a body that has a speed V1 sufficient to move in a circle around the Earth
There is only one force acting on the Earth's satellite - the force of gravity directed towards the center of the Earth
First escape velocity- this is the speed that must be imparted to the body so that it revolves around the planet in a circular orbit.

where R is the distance from the center of the planet to the satellite.
For the Earth, near its surface, the first escape velocity is equal to

1.3. Basic concepts and laws of statics and hydrostatics

A body (material point) is in a state of equilibrium if the vector sum of the forces acting on it is equal to zero. There are 3 types of equilibrium: stable, unstable and indifferent. If, when a body is removed from an equilibrium position, forces arise that tend to bring this body back, this stable balance. If forces arise that tend to move the body further from the equilibrium position, this unstable position; if no forces arise - indifferent(see Fig. 3).


When we are not talking about a material point, but about a body that can have an axis of rotation, then in order to achieve an equilibrium position, in addition to the equality of the sum of forces acting on the body to zero, it is necessary that the algebraic sum of the moments of all forces acting on the body be equal to zero.

Here d is the force arm. Shoulder of strength d is the distance from the axis of rotation to the line of action of the force.

Lever equilibrium condition:
the algebraic sum of the moments of all forces rotating the body is equal to zero.
Pressure is a physical quantity equal to the ratio of the force acting on a platform perpendicular to this force to the area of ​​the platform:

Valid for liquids and gases Pascal's law:
pressure spreads in all directions without changes.
If a liquid or gas is in a gravity field, then each layer above presses on the layers below, and as the liquid or gas is immersed inside, the pressure increases. For liquids

where ρ is the density of the liquid, h is the depth of penetration into the liquid.

A homogeneous liquid in communicating vessels is established at the same level. If liquid with different densities is poured into the elbows of communicating vessels, then the liquid with a higher density is installed at a lower height. In this case

The heights of liquid columns are inversely proportional to densities:

Hydraulic Press is a vessel filled with oil or other liquid, in which two holes are cut, closed by pistons. The pistons have different areas. If a certain force is applied to one piston, then the force applied to the second piston turns out to be different.
Thus, the hydraulic press serves to convert the magnitude of the force. Since the pressure under the pistons must be the same, then

Then A1 = A2.
A body immersed in a liquid or gas is acted upon by an upward buoyant force from the side of this liquid or gas, which is called by the power of Archimedes
The magnitude of the buoyancy force is determined by Archimedes' law: a body immersed in a liquid or gas is acted upon by a buoyant force directed vertically upward and equal to the weight of the liquid or gas displaced by the body:

where ρ liquid is the density of the liquid in which the body is immersed; V submergence is the volume of the submerged part of the body.

Body floating condition- a body floats in a liquid or gas when the buoyant force acting on the body is equal to the force of gravity acting on the body.

1.4. Conservation laws

Body impulse is a physical quantity equal to the product of a body’s mass and its speed:

Momentum is a vector quantity. [p] = kg m/s. Along with body impulse, they often use impulse of power. This is the product of force and the duration of its action
The change in the momentum of a body is equal to the momentum of the force acting on this body. For an isolated system of bodies (a system whose bodies interact only with each other) law of conservation of momentum: the sum of the impulses of the bodies of an isolated system before interaction is equal to the sum of the impulses of the same bodies after the interaction.
Mechanical work called a physical quantity that is equal to the product of the force acting on the body, the displacement of the body and the cosine of the angle between the direction of the force and the displacement:

Power is the work done per unit of time:

The ability of a body to do work is characterized by a quantity called energy. Mechanical energy is divided into kinetic and potential. If a body can do work due to its motion, it is said to have kinetic energy. The kinetic energy of the translational motion of a material point is calculated by the formula

If a body can do work by changing its position relative to other bodies or by changing the position of parts of the body, it has potential energy. An example of potential energy: a body raised above the ground, its energy is calculated using the formula

where h is the lift height

Compressed spring energy:

where k is the spring stiffness coefficient, x is the absolute deformation of the spring.

The sum of potential and kinetic energy is mechanical energy. For an isolated system of bodies in mechanics, law of conservation of mechanical energy: if there are no frictional forces between the bodies of an isolated system (or other forces leading to energy dissipation), then the sum of the mechanical energies of the bodies of this system does not change (the law of conservation of energy in mechanics). If there are friction forces between the bodies of an isolated system, then during interaction part of the mechanical energy of the bodies turns into internal energy.

1.5. Mechanical vibrations and waves

Oscillations movements that have varying degrees of repeatability over time are called. Oscillations are called periodic if the values ​​of physical quantities that change during the oscillation process are repeated at regular intervals.
Harmonic vibrations are called such oscillations in which the oscillating physical quantity x changes according to the law of sine or cosine, i.e.

The quantity A equal to the largest absolute value of the fluctuating physical quantity x is called amplitude of oscillations. The expression α = ωt + ϕ determines the value of x at a given time and is called the oscillation phase. Period T is the time it takes for an oscillating body to complete one complete oscillation. Frequency of periodic oscillations is the number of complete oscillations completed per unit of time:

Frequency is measured in s -1. This unit is called hertz (Hz).

Mathematical pendulum is a material point of mass m suspended on a weightless inextensible thread and oscillating in a vertical plane.
If one end of the spring is fixed motionless, and a body of mass m is attached to its other end, then when the body is removed from the equilibrium position, the spring will stretch and oscillations of the body on the spring will occur in the horizontal or vertical plane. Such a pendulum is called a spring pendulum.

Period of oscillation of a mathematical pendulum determined by the formula

where l is the length of the pendulum.

Period of oscillation of a load on a spring determined by the formula

where k is the spring stiffness, m is the mass of the load.

Propagation of vibrations in elastic media.
A medium is called elastic if there are interaction forces between its particles. Waves are the process of propagation of vibrations in elastic media.
The wave is called transverse, if the particles of the medium oscillate in directions perpendicular to the direction of propagation of the wave. The wave is called longitudinal, if the vibrations of the particles of the medium occur in the direction of wave propagation.
Wavelength is the distance between two closest points oscillating in the same phase:

where v is the speed of wave propagation.

Sound waves are called waves in which oscillations occur with frequencies from 20 to 20,000 Hz.
The speed of sound varies in different environments. The speed of sound in air is 340 m/s.
Ultrasonic waves are called waves whose oscillation frequency exceeds 20,000 Hz. Ultrasonic waves are not perceived by the human ear.