To be able to characterize the energy characteristics of motion, the concept of mechanical work was introduced. And it is to her in her various manifestations that the article is devoted. To understand the topic is both easy and quite complex. The author sincerely tried to make it more understandable and understandable, and one can only hope that the goal has been achieved.
What is mechanical work?
What is it called? If some force works on the body, and as a result of the action of this force, the body moves, then this is called mechanical work. When approached from the point of view of scientific philosophy, several additional aspects can be distinguished here, but the article will cover the topic from the point of view of physics. Mechanical work is not difficult if you think carefully about the words written here. But the word "mechanical" is usually not written, and everything is reduced to the word "work". But not every job is mechanical. Here a man sits and thinks. Does it work? Mentally yes! But is it mechanical work? No. What if the person is walking? If the body moves under the influence of a force, then this is mechanical work. Everything is simple. In other words, the force acting on the body does (mechanical) work. And one more thing: it is work that can characterize the result of the action of a certain force. So if a person walks, then certain forces (friction, gravity, etc.) perform mechanical work on a person, and as a result of their action, a person changes his point of location, in other words, he moves.
Work like physical quantity equals the force that acts on the body, multiplied by the path that the body made under the influence of this force and in the direction indicated by it. We can say that mechanical work was done if 2 conditions were simultaneously met: the force acted on the body, and it moved in the direction of its action. But it was not performed or is not performed if the force acted, and the body did not change its location in the coordinate system. Here are small examples where mechanical work is not done:
- So a person can fall on a huge boulder in order to move it, but there is not enough strength. The force acts on the stone, but it does not move, and work does not occur.
- The body moves in the coordinate system, and the force is equal to zero or they are all compensated. This can be observed during inertial motion.
- When the direction in which the body moves is perpendicular to the force. When the train moves along a horizontal line, the force of gravity does not do its work.
Depending on certain conditions, mechanical work can be negative and positive. So, if the directions and forces, and the movements of the body are the same, then positive work occurs. An example of positive work is the effect of gravity on a falling drop of water. But if the force and direction of movement are opposite, then negative mechanical work occurs. An example of such an option is a balloon rising up and gravity, which does negative work. When a body is subjected to the influence of several forces, such work is called "resultant force work".
Features of practical application (kinetic energy)
We pass from theory to practical part. Separately, we should talk about mechanical work and its use in physics. As many probably remembered, all the energy of the body is divided into kinetic and potential. When an object is in equilibrium and is not moving anywhere, it potential energy equals the total energy, and the kinetic equals zero. When the movement begins, the potential energy begins to decrease, the kinetic energy to increase, but in total they are equal to the total energy of the object. For a material point, kinetic energy is defined as the work of the force that accelerated the point from zero to the value H, and in formula form, the kinetics of the body is ½ * M * H, where M is the mass. To find out the kinetic energy of an object that consists of many particles, you need to find the sum of all the kinetic energy of the particles, and this will be the kinetic energy of the body.
Features of practical application (potential energy)
In the case when all the forces acting on the body are conservative, and the potential energy is equal to the total, then no work is done. This postulate is known as the law of conservation of mechanical energy. Mechanical energy in a closed system is constant in the time interval. The conservation law is widely used to solve problems from classical mechanics.
Features of practical application (thermodynamics)
In thermodynamics, the work done by a gas during expansion is calculated by the integral of pressure multiplied by volume. This approach is applicable not only in cases where there is an exact function of volume, but also to all processes that can be displayed in the pressure/volume plane. The knowledge of mechanical work is also applied not only to gases, but to everything that can exert pressure.
Features of practical application in practice (theoretical mechanics)
IN theoretical mechanics all the properties and formulas described above are considered in more detail, in particular, these are projections. She also gives her own definition for various formulas of mechanical work (an example of the definition for the Rimmer integral): the limit to which the sum of all the forces of elementary work tends when the fineness of the partition tends to zero is called the work of the force along the curve. Probably difficult? But nothing with theoretical mechanics All. Yes, and all the mechanical work, physics and other difficulties are over. Further there will be only examples and a conclusion.
Mechanical work units
The SI uses joules to measure work, while the GHS uses ergs:
- 1 J = 1 kg m²/s² = 1 Nm
- 1 erg = 1 g cm²/s² = 1 dyn cm
- 1 erg = 10 −7 J
Examples of mechanical work
In order to finally understand such a concept as mechanical work, you should study a few separate examples that will allow you to consider it from many, but not all, sides:
- When a person lifts a stone with his hands, then mechanical work occurs with the help of the muscular strength of the hands;
- When a train travels along the rails, it is pulled by the traction force of the tractor (electric locomotive, diesel locomotive, etc.);
- If you take a gun and shoot from it, then thanks to the pressure force that the powder gases will create, work will be done: the bullet is moved along the barrel of the gun at the same time as the speed of the bullet itself increases;
- There is also mechanical work when the friction force acts on the body, forcing it to reduce the speed of its movement;
- The above example with the balls as they rise in opposite side relative to the direction of the force of gravity, is also an example of mechanical work, but in addition to the force of gravity, the force of Archimedes also acts when everything lighter than air rises.
What is power?
Finally, I want to touch on the topic of power. The work done by a force in one unit of time is called power. In fact, power is such a physical quantity that is a reflection of the ratio of work to a certain period of time during which this work was done: M = P / B, where M is power, P is work, B is time. The SI unit of power is 1 watt. A watt is equal to the power that does the work of one joule in one second: 1 W = 1J \ 1s.
mechanical work- this is a physical quantity - a scalar quantitative measure of the action of a force (resultant force) on a body or forces on a system of bodies. Depends on the numerical value and direction of the force (forces) and on the displacement of the body (system of bodies).
Notation used
Work is usually denoted by the letter A(from him. A rbeit- work, labor) or a letter W(from English. w ork- work, work).
Definition
The work of a force applied to a material point
The total work to move one material point, performed by several forces applied to this point, is defined as the work of the resultant of these forces (their vector sum). Therefore, we will continue to talk about one force applied to a material point.
At rectilinear motion material point and constant value force applied to it, the work (of this force) is equal to the product of the projection of the force vector on the direction of movement and the length of the displacement vector made by the point:
A = F s s = F s c o s (F , s) = F → ⋅ s → (\displaystyle A=F_(s)s=Fs\ \mathrm (cos) (F,s)=(\vec (F))\ cdot(\vec(s))) A = ∫ F → ⋅ d s → . (\displaystyle A=\int (\vec (F))\cdot (\vec (ds)).)(summation along the curve is implied, which is the limit of a broken line made up of successive displacements d s → , (\displaystyle (\vec (ds)),) if we first consider them finite, and then let the length of each tend to zero).
If there is a dependence of the force on the coordinates, the integral is defined as follows:
A = ∫ r → 0 r → 1 F → (r →) ⋅ d r → (\displaystyle A=\int \limits _((\vec (r))_(0))^((\vec (r)) _(1))(\vec (F))\left((\vec (r))\right)\cdot (\vec (dr))),Where r → 0 (\displaystyle (\vec(r))_(0)) And r → 1 (\displaystyle (\vec(r))_(1)) are the radius vectors of the initial and final position of the body, respectively.
- Consequence. If the direction of the applied force is orthogonal to the displacement of the body or the displacement is zero, then the work (of this force) is zero.
The work of forces applied to a system of material points
The work of forces in moving a system of material points is defined as the sum of the work of these forces in moving each point (the work done on each point of the system is summed up in the work of these forces on the system).
Even if the body is not a system of discrete points, it can be divided (mentally) into many infinitesimal elements (pieces), each of which can be considered a material point, and work can be calculated in accordance with the definition above. In this case, the discrete sum is replaced by an integral.
- These definitions can be used both to calculate the work of a particular force or class of forces, and to calculate the total work done by all forces acting on the system.
Kinetic energy
E k = 1 2 m v 2 . (\displaystyle E_(k)=(\frac (1)(2))mv^(2).)For complex objects consisting of many particles, the kinetic energy of the body is equal to the sum of the kinetic energies of the particles.
Potential energy
Work in thermodynamics
In thermodynamics, the work done by a gas during expansion is calculated as the integral of pressure over volume:
A 1 → 2 = ∫ V 1 V 2 P d V . (\displaystyle A_(1\rightarrow 2)=\int \limits _(V_(1))^(V_(2))PdV.)
The work done on the gas coincides with this expression in absolute value, but is opposite in sign.
- The natural generalization of this formula is applicable not only to processes where pressure is a single-valued function of volume, but also to any process (depicted by any curve in the plane PV), in particular, to cyclic processes.
- In principle, the formula is applicable not only to gas, but also to anything capable of exerting pressure (it is only necessary that the pressure in the vessel be the same everywhere, which is implicitly implied in the formula).
This formula is directly related to mechanical work. Indeed, let's try to write the mechanical work during the expansion of the vessel, given that the gas pressure force will be directed perpendicular to each elementary area, equal to the product of pressure P To the square dS platforms, and then the work done by the gas to displace h one such elementary site will be
d A = P d S h . (\displaystyle dA=PdSh.)It can be seen that this is the product of pressure and volume increment near a given elementary area. And summing over all dS, we will get the final result, where there will already be a full increase in volume, as in the main formula of the section.
Work of force in theoretical mechanics
Let us consider in more detail than it was done above the construction of the definition of energy as a Riemannian integral.
Let the material point M (\displaystyle M) moves along a continuously differentiable curve G = ( r = r (s) ) (\displaystyle G=\(r=r(s)\)), where s is the variable length of the arc, 0 ≤ s ≤ S (\displaystyle 0\leq s\leq S), and it is acted upon by a force directed tangentially to the trajectory in the direction of motion (if the force is not directed tangentially, then we will understand by F (s) (\displaystyle F(s)) projection of the force onto the positive tangent of the curve, thus reducing this case to the one considered below). Value F (ξ i) △ s i , △ s i = s i − s i − 1 , i = 1 , 2 , . . . , i τ (\displaystyle F(\xi _(i))\triangle s_(i),\triangle s_(i)=s_(i)-s_(i-1),i=1,2,... ,i_(\tau )), is called elementary work strength F (\displaystyle F) on the site and is taken as an approximate value of the work that the force produces F (\displaystyle F) acting on the material point when the latter passes the curve G i (\displaystyle G_(i)). The sum of all elementary works is the Riemann integral sum of the function F (s) (\displaystyle F(s)).
In accordance with the definition of the Riemann integral, we can define work:
The limit to which the sum tends ∑ i = 1 i τ F (ξ i) △ s i (\displaystyle \sum _(i=1)^(i_(\tau ))F(\xi _(i))\triangle s_(i)) of all elementary works, when pettiness | τ | (\displaystyle |\tau |) partitions τ (\displaystyle \tau ) tends to zero is called the work done by the force. F (\displaystyle F) along the curve G (\displaystyle G).
Thus, if we denote this work by the letter W (\displaystyle W), then, due to this definition,
W = lim | τ | → 0 ∑ i = 1 i τ F (ξ i) △ s i (\displaystyle W=\lim _(|\tau |\rightarrow 0)\sum _(i=1)^(i_(\tau ))F( \xi _(i))\triangle s_(i)),hence,
W = ∫ 0 s F (s) d s (\displaystyle W=\int \limits _(0)^(s)F(s)ds) (1).If the position of a point on the trajectory of its movement is described using some other parameter t (\displaystyle t)(for example, time) and if the distance traveled s = s (t) (\displaystyle s=s(t)), a ≤ t ≤ b (\displaystyle a\leq t\leq b) is a continuously differentiable function, then from formula (1) we obtain
W = ∫ a b F [ s (t) ] s ′ (t) d t . (\displaystyle W=\int \limits _(a)^(b)Fs"(t)dt.)Dimension and units
The unit of measure for work in the International System of Units (SI) is
What does it mean?
In physics, "mechanical work" is the work of some force (gravity, elasticity, friction, etc.) on the body, as a result of which the body moves.
Often the word "mechanical" is simply not spelled.
Sometimes you can find the expression "the body has done the work", which basically means "the force acting on the body has done the work."
I think - I'm working.
I go - I also work.
Where is the mechanical work here?
If a body moves under the action of a force, then mechanical work is done.
The body is said to do work.
More precisely, it will be like this: the work is done by the force acting on the body.
Work characterizes the result of the action of a force.
The forces acting on a person do mechanical work on him, and as a result of the action of these forces, the person moves.
Work is a physical quantity equal to the product of the force acting on the body and the path taken by the body under the action of the force in the direction of this force.
A - mechanical work,
F - strength,
S - the distance traveled.
Work is done, if 2 conditions are met simultaneously: a force acts on the body and it
moves in the direction of the force.
Work is not done(i.e. equal to 0) if:
1. The force acts, but the body does not move.
For example: we act with force on a stone, but we cannot move it.
2. The body moves, and the force is equal to zero, or all forces are compensated (ie, the resultant of these forces is equal to 0).
For example: when moving by inertia, no work is done.
3. The direction of the force and the direction of motion of the body are mutually perpendicular.
For example: when a train moves horizontally, gravity does no work.
Work can be positive or negative.
1. If the direction of the force and the direction of motion of the body are the same, positive work is done.
For example: gravity, acting on a drop of water falling down, does positive work.
2. If the direction of the force and the movement of the body are opposite, negative work is done.
For example: the force of gravity acting on a rising balloon does negative work.
If several forces act on a body, then the total work of all forces is equal to the work of the resulting force.
Units of work
In honor of the English scientist D. Joule, the unit of work was named 1 Joule.
In the international system of units (SI):
[A] = J = N m
1J = 1N 1m
Mechanical work is equal to 1 J if, under the influence of a force of 1 N, the body moves 1 m in the direction of this force.
When flying from the thumb of a person to the index
a mosquito does work - 0,000,000,000,000,000,000,000,000,001 J.
The human heart performs approximately 1 J of work in one contraction, which corresponds to the work done when lifting a load of 10 kg to a height of 1 cm.
TO WORK, FRIENDS!
Definition
In the event that under the influence of a force there is a change in the modulus of the velocity of the body, then they say that the force makes work. It is believed that if the speed increases, then the work is positive, if the speed decreases, then the work done by the force is negative. The change in the kinetic energy of a material point in the course of its movement between two positions is equal to the work done by the force:
The action of a force on a material point can be characterized not only by changing the speed of the body, but by using the magnitude of the displacement that the body in question makes under the action of force ().
elementary work
The elementary work of some force is defined as the scalar product:
Radius is the vector of the point to which the force is applied, is the elementary movement of the point along the trajectory, is the angle between the vectors and . If the work is an obtuse angle, the work is less than zero, if the angle is acute, then the work is positive, with
In Cartesian coordinates, formula (2) has the form:
where F x ,F y ,F z are vector projections onto Cartesian axes.
When considering the work of a force applied to a material point, you can use the formula:
where is the velocity of the material point, is the momentum of the material point.
If on the body ( mechanical system) several forces act simultaneously, then the elementary work that these forces perform on the system is equal to:
where the summation of the elementary work of all forces is carried out, dt is a small period of time during which elementary work is performed on the system.
The resulting work of internal forces, even if the rigid body is moving, is zero.
Let a rigid body rotate around a fixed point - the origin of coordinates (or a fixed axis that passes through this point). In this case, the elementary work of all external forces(assuming that their number is n) that act on the body is equal to:
where is the resulting moment of forces relative to the point of rotation, is the elementary rotation vector, and is the instantaneous angular velocity.
The work of the force on the final section of the trajectory
If the force does work to move the body in the final section of the trajectory of its movement, then the work can be found as:
In the event that the force vector is a constant value over the entire segment of the movement, then:
where is the projection of the force on the tangent to the trajectory.
Work units
The basic unit of measurement of the moment of work in the SI system is: [A] \u003d J \u003d N m
In CGS: [A]=erg=dyn cm
1J=10 7 erg
Examples of problem solving
Example
Exercise. Material point moves in a straight line (Fig. 1) under the influence of a force, which is given by the equation: . The force is directed along the motion of the material point. What is the work of this force on the segment of the path from s=0 to s=s 0 ?
Solution. As a basis for solving the problem, we take the formula for calculating the work of the form:
where , the same as according to the condition of the problem . We substitute the expression for the force modulus given by the conditions, take the integral:
Answer.
Example
Exercise. The material point moves in a circle. Its speed changes according to the expression: . In this case, the work of the force that acts on a point is proportional to time: . What is the value of n?
« Physics - Grade 10 "
The law of conservation of energy is a fundamental law of nature that allows describing most of the phenomena that occur.
The description of the motion of bodies is also possible with the help of such concepts of dynamics as work and energy.
Remember what work and power are in physics.
Do these concepts coincide with everyday ideas about them?
All our daily actions boil down to the fact that with the help of muscles we either set the surrounding bodies in motion and maintain this movement, or we stop the moving bodies.
These bodies are tools (hammer, pen, saw), in games - balls, pucks, chess pieces. In production and agriculture people also set tools in motion.
The use of machines greatly increases labor productivity due to the use of engines in them.
The purpose of any engine is to set the bodies in motion and maintain this movement, despite braking by both ordinary friction and “working” resistance (the cutter must not only slide over the metal, but, crashing into it, remove chips; the plow must loosen land, etc.). In this case, a force must act on the moving body from the side of the engine.
Work is always done in nature when a force (or several forces) from another body (other bodies) acts on a body in the direction of its movement or against it.
The gravitational force does work when rain drops or a stone fall from a cliff. At the same time, the work is done by the resistance force acting on the falling drops or on the stone from the side of the air. The elastic force also does work when a tree bent by the wind straightens.
Job definition.
Newton's second law in impulsive form ∆=∆t allows you to determine how the speed of the body changes in absolute value and direction, if a force acts on it during the time Δt.
The impact on bodies of forces, leading to a change in the modulus of their velocity, is characterized by a value that depends both on the forces and on the displacements of the bodies. This quantity in mechanics is called work of force.
Modulo change of speed is possible only when the projection of the force F r on the direction of body movement is nonzero. It is this projection that determines the action of the force that changes the velocity of the body modulo. She does the work. Therefore, the work can be considered as the product of the projection of the force F r by the displacement modulus |Δ| (Fig. 5.1):
А = F r |Δ|. (5.1)
If the angle between force and displacement is denoted by α, then F r = Fcosα.
Therefore, the work is equal to:
A = |Δ|cosα. (5.2)
Our everyday concept of work differs from the definition of work in physics. You are holding a heavy suitcase, and it seems to you that you are doing work. However, from the point of view of physics, your work is equal to zero.
The work of a constant force is equal to the product of the modules of force and the displacement of the point of application of the force and the cosine of the angle between them.
In general, when moving solid body move it different points are different, but when determining the work of a force, we Δ understand the movement of its point of application. In the translational motion of a rigid body, the displacement of all its points coincides with the displacement of the point of application of the force.
Work, unlike force and displacement, is not a vector, but a scalar quantity. It can be positive, negative or zero.
The sign of work is determined by the sign of the cosine of the angle between force and displacement. If α< 90°, то А >0 since the cosine sharp corners positive. For α > 90°, the work is negative, since the cosine of obtuse angles is negative. At α = 90° (the force is perpendicular to the displacement), no work is done.
If several forces act on the body, then the projection of the resultant force on the displacement is equal to the sum of the projections of the individual forces:
F r = F 1r + F 2r + ... .
Therefore, for the work of the resultant force, we obtain
A = F 1r |Δ| + F 2r |Δ| + ... = A 1 + A 2 + .... (5.3)
If several forces act on the body, then the total work ( algebraic sum work of all forces) is equal to the work of the resultant force.
The work done by force can be represented graphically. Let us explain this by depicting in the figure the dependence of the projection of the force on the coordinate of the body when it moves in a straight line.
Let the body move along the OX axis (Fig. 5.2), then
Fcosα = F x , |Δ| = Δ x.
For the work of the force, we get
А = F|Δ|cosα = F x Δx.
Obviously, the area of the rectangle shaded in Figure (5.3, a) is numerically equal to the work done when the body moves from a point with coordinate x1 to a point with coordinate x2.
Formula (5.1) is valid when the projection of the force on the displacement is constant. In the case of a curved trajectory, constant or variable force, we divide the trajectory into small segments, which can be considered rectilinear, and the projection of the force on a small displacement Δ - permanent.
Then, calculating the work done on each displacement Δ
and then summing up these works, we determine the work of the force on the final displacement (Fig. 5.3, b). Unit of work.
The unit of work can be set using the basic formula (5.2). If, when a body moves per unit length, a force acts on it, the modulus of which equal to one, and the direction of the force coincides with the direction of movement of its point of application (α = 0), then the work will be equal to one. IN international system(SI) unit of work is the joule (denoted J):
1 J = 1 N 1 m = 1 N m.
Joule is the work done by a force of 1 N at a displacement of 1 if the directions of the force and displacement coincide.
Multiple units of work are often used - kilojoule and mega joule:
1 kJ = 1000 J,
1 MJ = 1000000 J.
Work can be done in a long period of time, or in a very small one. In practice, however, it is far from indifferent whether work can be done quickly or slowly. The time during which work is done determines the performance of any engine. A tiny electric motor can do a lot of work, but it will take a lot of time. Therefore, along with work, a value is introduced that characterizes the speed with which it is produced - power.
Power is the ratio of work A to the time interval Δt for which this work is done, i.e. power is the rate of work:
Substituting in formula (5.4) instead of work A its expression (5.2), we obtain
Thus, if the force and speed of the body are constant, then the power is equal to the product of the modulus of the force vector by the modulus of the velocity vector and the cosine of the angle between the directions of these vectors. If these quantities are variables, then by formula (5.4) one can determine the average power similarly to the definition average speed body movements.
The concept of power is introduced to evaluate the work per unit of time performed by some mechanism (pump, crane, machine motor, etc.). Therefore, in formulas (5.4) and (5.5), by always means the thrust force.
In SI, power is expressed in terms of watts (W).
The power is 1 W if the work equal to 1 J is done in 1 s.
Along with the watt, larger (multiple) units of power are used:
1 kW (kilowatt) = 1000 W,
1 MW (megawatt) = 1,000,000 W.
