What does mechanical work depend on? When solving problems, you need to remember that. Work done by resultant force

If a force acts on a body, then this force does work to move this body. Before defining work at curvilinear movement material point, consider special cases:

In this case the mechanical work A is equal to:

A= F scos=
,

or A = Fcos× s = F S × s,

WhereF S – projection strength to move. In this case F s = const, And geometric meaning work A is the area of ​​the rectangle constructed in coordinates F S , , s.

Let's plot the projection of force on the direction of movement F S as a function of displacement s. Let us represent the total displacement as the sum of n small displacements
. For small i -th movement
work is equal

or the area of ​​the shaded trapezoid in the figure.

.

The value under the integral will represent the elementary work of infinitesimal displacement
:

- basic work.

–work in curved motion.

Example 1: Work of gravity
during curvilinear motion of a material point.

Further as a constant value can be taken out of the integral sign, and the integral according to the figure will represent the full displacement . .

If we denote the height of a point 1 from the Earth's surface through , and the height of the point 2 through , That

We see that in this case the work is determined by the position of the material point at the initial and final moments of time and does not depend on the shape of the trajectory or path. The work done by gravity along a closed path is zero:
.

Forces whose work on a closed path is zero are calledconservative .

Example 2 : Work done by friction force.

This is an example of a non-conservative force. To show this, it is enough to consider the elementary work of the friction force:

,

those. The work done by the friction force is always a negative quantity and cannot be equal to zero on a closed path. The work done per unit time is called power. If during the time
work is being done
, then the power is equal

–mechanical power.

Taking
as

,

we get the expression for power:

.

The SI unit of work is the joule:
= 1 J = 1 N 1 m, and the unit of power is the watt: 1 W = 1 J/s.

Mechanical energy.

Energy is a general quantitative measure of the movement of interaction of all types of matter. Energy does not disappear and does not arise from nothing: it can only pass from one form to another. The concept of energy links together all phenomena in nature. In accordance with the various forms of motion of matter, different types of energy are considered - mechanical, internal, electromagnetic, nuclear, etc.

The concepts of energy and work are closely related to each other. It is known that work is done due to the energy reserve and, conversely, by doing work, it is possible to increase the energy reserve in any device. In other words, work is a quantitative measure of energy change:

.

Energy, like work in SI, is measured in joules: [ E]=1 J.

Mechanical energy is of two types - kinetic and potential.

Kinetic energy (or energy of motion) is determined by the masses and velocities of the bodies in question. Let's consider material point, moving under the influence of force . The work of this force increases the kinetic energy of a material point
. In this case, let us calculate the small increment (differential) of kinetic energy:

When calculating
Newton's second law was used
, and
- module of the velocity of the material point. Then
can be represented as:

-

- kinetic energy of a moving material point.

Multiplying and dividing this expression by
, and given that
, we get

-

- connection between momentum and kinetic energy of a moving material point.

Potential energy ( or the energy of the position of bodies) is determined by the action of conservative forces on the body and depends only on the position of the body .

We have seen that the work done by gravity
with curvilinear motion of a material point
can be represented as the difference in function values
, taken at the point 1 and at the point 2 :

.

It turns out that whenever the forces are conservative, the work of these forces on the path 1
2 can be represented as:

.

Function , which depends only on the position of the body is called potential energy.

Then for elementary work we get

–work equals loss of potential energy.

Otherwise, we can say that work is done due to the reserve of potential energy.

Size , equal to the sum of the kinetic and potential energies of the particle, is called the total mechanical energy of the body:

–full mechanical energy body.

In conclusion, we note that using Newton’s second law
, kinetic energy differential
can be represented as:

.

Potential energy differential
, as indicated above, is equal to:

.

Thus, if the force – conservative force and no others external forces, That , i.e. in this case, the total mechanical energy of the body is conserved.

The horse pulls the cart with some force, let's denote it F traction. Grandfather, sitting on the cart, presses on it with some force. Let's denote it F pressure The cart moves along the direction of the horse's traction force (to the right), but in the direction of the grandfather's pressure force (downward) the cart does not move. That's why in physics they say that F traction does work on the cart, and F the pressure does not do work on the cart.

So, work of force on the body or mechanical work– a physical quantity whose modulus is equal to the product of the force and the path traveled by the body along the direction of action of this force s:

In honor of the English scientist D. Joule, the unit of mechanical work was named 1 joule(according to the formula, 1 J = 1 N m).

If a certain force acts on the body in question, then some body acts on it. That's why the work of force on the body and the work of the body on the body are complete synonyms. However, the work of the first body on the second and the work of the second body on the first are partial synonyms, since the moduli of these works are always equal, and their signs are always opposite. That is why there is a “±” sign in the formula. Let's discuss the signs of work in more detail.

Numerical values ​​of force and path are always non-negative quantities. In contrast, mechanical work can have both positive and negative signs. If the direction of the force coincides with the direction of motion of the body, then the work done by the force is considered positive. If the direction of force is opposite to the direction of motion of the body, the work done by a force is considered negative(we take “–” from the “±” formula). If the direction of motion of the body is perpendicular to the direction of the force, then such a force does not do any work, that is, A = 0.

Consider three illustrations of three aspects of mechanical work.

Doing work by force may look different from the perspective of different observers. Let's consider an example: a girl rides up in an elevator. Does it perform mechanical work? A girl can do work only on those bodies that are acted upon by force. There is only one such body - the elevator cabin, since the girl presses on its floor with her weight. Now we need to find out whether the cabin goes a certain way. Let's consider two options: with a stationary and a moving observer.

Let the observer boy sit on the ground first. In relation to it, the elevator car moves upward and passes a certain distance. The girl’s weight is directed in the opposite direction - down, therefore, the girl performs negative mechanical work on the cabin: A dev< 0. Вообразим, что мальчик-наблюдатель пересел внутрь кабины движущегося лифта. Как и ранее, вес девочки действует на пол кабины. Но теперь по отношению к такому наблюдателю кабина лифта не движется. Поэтому с точки зрения наблюдателя в кабине лифта девочка не совершает механическую работу: A dev = 0.

Mechanical work This is an energy characteristic of the movement of physical bodies, which has a scalar form. It is equal to the modulus of the force acting on the body, multiplied by the modulus of the displacement caused by this force and by the cosine of the angle between them.

Formula 1 - Mechanical work.

F - Force acting on the body.

s - Body movement.

cosa - Cosine of the angle between force and displacement.

This formula has general form. If the angle between the applied force and the displacement is zero, then the cosine is equal to 1. Accordingly, the work will be equal only to the product of the force and the displacement. Simply put, if a body moves in the direction of application of force, then mechanical work is equal to the product of force and displacement.

Second special case, when the angle between the force acting on the body and its displacement is 90 degrees. In this case, the cosine of 90 degrees is equal to zero, so the work will be equal to zero. And indeed, what happens is that we apply force in one direction, and the body moves perpendicular to it. That is, the body clearly does not move under the influence of our force. Thus, the work done by our force to move the body is zero.

Figure 1 - Work of forces when moving a body.

If more than one force acts on a body, then the total force acting on the body is calculated. And then it is substituted into the formula as the only force. A body under the influence of force can move not only rectilinearly, but also along an arbitrary trajectory. In this case, the work is calculated for a small section of movement, which can be considered rectilinear, and then summed up along the entire path.

Work can be both positive and negative. That is, if the displacement and force coincide in direction, then the work is positive. And if a force is applied in one direction, and the body moves in another, then the work will be negative. An example of negative work is the work of a frictional force. Since the friction force is directed counter to the movement. Imagine a body moving along a plane. A force applied to a body pushes it in a certain direction. This force makes positive work by body movement. But at the same time, the friction force does negative work. It slows down the movement of the body and is directed towards its movement.

Figure 2 - Force of motion and friction.

Mechanical work is measured in Joules. One Joule is the work done by a force of one Newton when moving a body one meter. In addition to the direction of movement of the body, the magnitude of the applied force can also change. For example, when a spring is compressed, the force applied to it will increase in proportion to the distance traveled. In this case, the work is calculated using the formula.

Formula 2 - Work of compression of a spring.

k is the spring stiffness.

x - moving coordinate.

One of the most important concepts in mechanics is work of force .

Work of force

All physical bodies in the world around us are set in motion with the help of force. If a moving body in the same or opposite direction is acted upon by a force or several forces from one or more bodies, then it is said that work is being done .

That is, mechanical work is performed by a force acting on the body. Thus, the traction force of an electric locomotive sets the entire train in motion, thereby performing mechanical work. The bicycle is driven by the muscular power of the cyclist's legs. Consequently, this force also does mechanical work.

In physics work of force call a physical quantity equal to the product of the force modulus, the displacement modulus of the point of application of the force and the cosine of the angle between the force and displacement vectors.

A = F s cos (F, s) ,

Where F force module,

s – travel module .

Work is always done if the angle between the winds of force and displacement is not zero. If the force acts in the direction opposite to the direction of motion, the amount of work is negative.

No work is done if no forces act on the body, or if the angle between the applied force and the direction of movement is 90 o (cos 90 o = 0).

If a horse pulls a cart, then the horse's muscular force, or the traction force directed along the direction of the cart's movement, does work. But the force of gravity with which the driver presses on the cart does not do any work, since it is directed downward, perpendicular to the direction of movement.

The work of force is a scalar quantity.

Unit of work in the SI measurement system - joule. 1 joule is the work done by a force of 1 newton at a distance of 1 m if the directions of the force and displacement coincide.

If several forces act on a body or a material point, then we speak of the work done by their resultant force.

If the applied force is not constant, then its work is calculated as an integral:

Power

The force that moves a body does mechanical work. But how this work is done, quickly or slowly, is sometimes very important to know in practice. After all, the same work can be done in different time. The work that a large electric motor does can be done by a small motor. But he will need much more time for this.

In mechanics, there is a quantity that characterizes the speed of work. This quantity is called power.

Power is the ratio of work performed in a certain period of time to the value of this period.

N= A /∆ t

A-priory A = F s cos α , A s/∆ t = v , hence

N= F v cos α = F v ,

Where F - force, v speed, α – the angle between the direction of force and the direction of speed.

That is power – this is the scalar product of the force vector and the velocity vector of the body.

In the international SI system, power is measured in watts (W).

1 watt of power is 1 joule (J) of work done in 1 second (s).

Power can be increased by increasing the force doing work or the rate at which this work is done.

Energy - universal measure various forms of movement and interaction. A change in the mechanical movement of the body is caused by forces, acting on it from other bodies. Works of force - the process of energy exchange between interacting bodies.

If there is movement on the body straight forward there is a constant force F acting, which makes a certain angle  with the direction of movement, then the work of this force is equal to the product of the projection of the force F s by the direction of movement multiplied by the movement of the point of application of the force: (1)

In general, the force can change both in magnitude and direction, therefore scalar value e elementary work forces F on displacement dr:

where  is the angle between vectors F and dr; ds = |dr| - elementary path; F s - projection of vector F onto vector dr Fig. 1

Work of force on the trajectory section from the point 1 to the point 2 equal to the algebraic sum of elementary work on individual infinitesimal sections of the path: (2)

Where s- passed through the body. When </2 работа силы положительна, если >/2 the work done by the force is negative. When =/2 (the force is perpendicular to the displacement), the work done by the force is zero.

Unit of work - joule(J): work done by a force of 1 N along a path of 1 m (1 J = 1 N  m).

Power– value of the speed of work: (3)

During time d t force F does work Fdr, and the power developed by this force at a given moment of the belt: (4)

i.e., it is equal to the scalar product of the force vector and the velocity vector with which the point of application of this force moves; N- magnitude scalar.

Unit of power - watt(W): power at which 1J of work is performed in 1s (1W = 1J/s).

Kinetic and potential energy

Kinetic energy mechanical system - the energy of mechanical movement of this system.

Force F, acting on a body at rest and causing it to move, does work, and the energy of the moving body changes (d T) increases by the amount of work expended d A. That is, dA = dT

Using Newton's second law (F=mdV/dt) and a number of other transformations we obtain

(5) - kinetic energy of a body of mass m moving at speed v.

Kinetic energy depends only on the mass and speed of the body.

In different inertial systems references moving relative to each other, the speed of the body, and therefore its kinetic energy, will not be the same. Thus, kinetic energy depends on the choice of reference frame.

Potential energy- mechanical energy of a system of bodies, determined by their relative position and the nature of the interaction forces between them.

When the bodies interact through force fields (fields of elastic, gravitational forces), the work done by the acting forces when moving the body does not depend on the trajectory of this movement, but depends only on the initial and final positions of the body. Such fields are called potential, and the forces acting in them are conservative. If the work done by a force depends on the trajectory of the body moving from one point to another, then such a force is called dissipative(friction force). A body, being in a potential field of forces, has potential energy P. The work of conservative forces with an elementary (infinitesimal) change in the configuration of the system is equal to the increment of potential energy taken with a minus sign: dA = - dP (6)

Work d A- the scalar product of force F and displacement dr and expression (6) can be written: Fdr= -dП (7)

When calculating, the potential energy of a body in a certain position is considered equal to zero (the zero reference level is chosen), and the energy of the body in other positions is measured relative to zero level.

The specific form of the function P depends on the nature of the force field. For example, the potential energy of a body of mass T, raised to a height h above the Earth's surface is equal to (8)

where is the height h is counted from the zero level, for which P 0 =0.

Since the origin is chosen arbitrarily, then potential energy may have a negative value (kinetic energy is always positive!). If we take the potential energy of a body lying on the surface of the Earth as zero, then the potential energy of a body located at the bottom of the mine (depth h" ), P= - mgh".

The potential energy of a system is a function of the state of the system. It depends only on the configuration of the system and its position in relation to external bodies.

Total mechanical energy of the system equal to the sum of kinetic and potential energies: E=T+P.