The work of the force is equal to the formula. mechanical work

Energy- a universal measure of various forms of movement and interaction. The change in the mechanical motion of the body is caused forces acting on it from other bodies. Power works - the process of energy exchange between interacting bodies.

If moving on the body straightforward a constant force F acts, 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 force application point: (1)

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

where  is the angle between the vectors F and dr; ds = |dr| - elementary way; F s - projection of the vector F onto the vector dr fig. one

The work of the force on the trajectory section from the point 1 to the point 2 is equal to the algebraic sum of elementary works on separate infinitesimal sections of the path: (2)

where s- passed by 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 of the force is zero.

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

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

During the time d t strength F does the work Fdr, and the power developed by this force is this moment 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.

Power unit - watt(W): power at which 1J work is done in 1s (1W = 1J/s).

Kinetic and potential energies

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

The force F, acting on a body at rest and causing its movement, does work, and the energy change of the moving body (d T) increases by the amount of work expended d A. i.e. 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 a speed v.

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

in different inertial systems reference, moving relative to each other, the speed of the body, and hence its kinetic energy will be different. Thus, the kinetic energy depends on the choice of the frame of reference.

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

In the case of the interaction of bodies carried out by means of 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 - conservative. If the work done by the force depends on the trajectory of the movement of the body from one point to another, then such a force is called dissipative(friction force). The body, being in a potential field of forces, has a potential energy P. The work of conservative forces with an elementary (infinitely small) change in the configuration of the system is equal to the increment of potential energy, taken with a minus sign: dA= - dП (6)

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

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

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

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

Since the origin is chosen arbitrarily, the potential energy can have a negative value (kinetic energy is always positive!). If we take as zero the potential energy of a body lying on the surface of the Earth, 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 is equal to the sum of kinetic and potential energies: E=T+P.

One of the most important concepts in mechanics work force .

Force work

All physical bodies in the world around us are driven by force. If a moving body in the same or opposite direction is affected by a force or several forces from one or more bodies, then they say that work is done .

That is, mechanical work is done by the 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 propelled by the muscular strength of the cyclist's legs. Therefore, this force also does mechanical work.

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

A = F s cos (F, s) ,

where F modulus of force,

s- movement module .

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

Work is not done if no forces act on the body, or if the angle between the applied force and the direction of motion is 90 o (cos 90 o \u003d 0).

If the horse pulls the cart, then the muscular force of the horse, or the traction force directed in the direction of the cart, does the work. And the force of gravity, with which the driver presses on the cart, does no work, since it is directed downward, perpendicular to the direction of movement.

The work of a force is a scalar quantity.

SI unit of work - joule. 1 joule is the work done by a force of 1 newton at a distance of 1 m if the direction of force and displacement are the same.

If on the body or material point Several forces act, then they talk about 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 sets the body in motion does mechanical work. But how this work is done, quickly or slowly, is sometimes very important to know in practice. For the same work can be done in different time. The work that a large electric motor does can be done by small motor. But it will take him much longer to do so.

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

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

N= A /∆ t

By definition A = F s cos α , but s/∆ t = v , Consequently

N= F v cos α = F v ,

where F - strength, v speed, α is the angle between the direction of the force and the direction of the velocity.

I.e power - is the scalar product of the force vector and the velocity vector of the body.

IN international system SI power is measured in watts (W).

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

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

Basic theoretical information

mechanical work

The energy characteristics of motion are introduced on the basis of the concept mechanical work or work force. Work done by a constant force F, is a physical quantity equal to the product of the modules of force and displacement, multiplied by the cosine of the angle between the force vectors F and displacement S:

Work is scalar value. It can be either positive (0° ≤ α < 90°), так и отрицательна (90° < α ≤ 180°). At α = 90° the work done by the force is zero. In the SI system, work is measured in joules (J). A joule is equal to the work done by a force of 1 newton to move 1 meter in the direction of the force.

If the force changes over time, then to find the work, they build a graph of the dependence of the force on the displacement and find the area of \u200b\u200bthe figure under the graph - this is the work:

An example of a force whose modulus depends on the coordinate (displacement) is the elastic force of a spring, which obeys Hooke's law ( F extr = kx).

Power

The work done by a force per unit of time is called power. Power P(sometimes referred to as N) is a physical quantity equal to the ratio of work A to the time span t during which this work was completed:

This formula calculates average power, i.e. power generally characterizing the process. So, work can also be expressed in terms of power: A = Pt(unless, of course, the power and time of doing the work are known). The unit of power is called the watt (W) or 1 joule per second. If the motion is uniform, then:

With this formula, we can calculate instant power(power at a given time), if instead of speed we substitute the value of instantaneous speed into the formula. How to know what power to count? If the task asks for power at a point in time or at some point in space, then it is considered instantaneous. If you are asking about power over a certain period of time or a section of the path, then look for the average power.

Efficiency - efficiency factor, is equal to the ratio of useful work to spent, or useful power to spent:

What work is useful and what is spent is determined from the condition specific task through logical reasoning. For example, if crane performs the work of lifting the load to a certain height, then the work of lifting the load will be useful (since it was for the sake of it that the crane was created), and the work done by the crane electric motor will be spent.

So, useful and expended power do not have a strict definition, and are found by logical reasoning. In each task, we ourselves must determine what in this task was the goal of doing the work ( useful work or power), and what was the mechanism or method of doing all the work (the expended power or work).

In the general case, the efficiency shows how efficiently the mechanism converts one type of energy into another. If the power changes over time, then the work is found as the area of the figure under the graph of power versus time:

Kinetic energy

A physical quantity equal to half the product of the body's mass and the square of its speed is called kinetic energy of the body (energy of motion):

That is, if a car with a mass of 2000 kg moves at a speed of 10 m/s, then it has a kinetic energy equal to E k \u003d 100 kJ and is capable of doing work of 100 kJ. This energy can turn into heat (when the car brakes, the rubber of the wheels, the road and brake discs) or can be spent to deform the car and the body that the car collided with (in an accident). When calculating kinetic energy, it does not matter where the car is moving, since energy, like work, is a scalar quantity.

A body has energy if it can do work. For example, a moving body has kinetic energy, i.e. the energy of motion, and is capable of doing work to deform bodies or impart acceleration to bodies with which a collision occurs.

physical meaning kinetic energy: in order for a body at rest with mass m began to move at a speed v it is necessary to do work equal to the obtained value of kinetic energy. If the body mass m moving at a speed v, then to stop it, it is necessary to do work equal to its initial kinetic energy. During braking, the kinetic energy is mainly (except for cases of collision, when the energy is used for deformation) “taken away” by the friction force.

Kinetic energy theorem: the work of the resultant force is equal to the change in the kinetic energy of the body:

The kinetic energy theorem is also valid in the general case when the body moves under the action of a changing force, the direction of which does not coincide with the direction of movement. It is convenient to apply this theorem in problems of acceleration and deceleration of a body.

Potential energy

Along with the kinetic energy or the energy of motion in physics, an important role is played by the concept potential energy or energy of interaction of bodies.

Potential energy is determined by the mutual position of the bodies (for example, the position of the body relative to the Earth's surface). The concept of potential energy can be introduced only for forces whose work does not depend on the trajectory of the body and is determined only by the initial and final positions (the so-called conservative forces). The work of such forces on a closed trajectory is zero. This property is possessed by the force of gravity and the force of elasticity. For these forces, we can introduce the concept of potential energy.

Potential energy of a body in the Earth's gravity field calculated by the formula:

The physical meaning of the potential energy of a body: potential energy is equal to the work done by gravity when lowering the body to zero level (h is the distance from the center of gravity of the body to the zero level). If a body has potential energy, then it is capable of doing work when this body falls from a height h down to zero. The work of gravity is equal to the change in the potential energy of the body, taken with the opposite sign:

Often in tasks for energy, you have to find work to lift (turn over, get out of the pit) the body. In all these cases, it is necessary to consider the movement not of the body itself, but only of its center of gravity.

The potential energy Ep depends on the choice of the zero level, that is, on the choice of the origin of the OY axis. In each problem, the zero level is chosen for reasons of convenience. It is not the potential energy itself that has physical meaning, but its change when the body moves from one position to another. This change does not depend on the choice of the zero level.

Potential energy of a stretched spring calculated by the formula:

where: k- spring stiffness. A stretched (or compressed) spring is capable of setting in motion a body attached to it, that is, imparting kinetic energy to this body. Therefore, such a spring has a reserve of energy. Stretch or Compression X must be calculated from the undeformed state of the body.

The potential energy of an elastically deformed body is equal to the work of the elastic force during the transition from a given state to a state with zero deformation. If in the initial state the spring was already deformed, and its elongation was equal to x 1 , then upon transition to a new state with elongation x 2, the elastic force will do work equal to the change in potential energy, taken with the opposite sign (since the elastic force is always directed against the deformation of the body):

Potential energy during elastic deformation is the energy of interaction of individual parts of the body with each other by elastic forces.

The work of the friction force depends on the distance traveled (this type of force whose work depends on the trajectory and the distance traveled is called: dissipative forces). The concept of potential energy for the friction force cannot be introduced.

Efficiency

Efficiency factor (COP)- a characteristic of the efficiency of a system (device, machine) in relation to the conversion or transfer of energy. It is determined by the ratio of useful energy used to the total amount of energy received by the system (the formula has already been given above).

Efficiency can be calculated both in terms of work and in terms of power. Useful and expended work (power) is always determined by simple logical reasoning.

IN electric motors Efficiency - the ratio of the performed (useful) mechanical work to electrical energy received from the source. In heat engines, the ratio of useful mechanical work to the amount of heat expended. IN electrical transformers- attitude electromagnetic energy received in the secondary winding to the energy consumed by the primary winding.

Due to its generality, the concept of efficiency makes it possible to compare and evaluate from a unified point of view such various systems, such as nuclear reactors, electric generators and engines, thermal power plants, semiconductor devices, biological objects, etc.

Due to the inevitable energy losses due to friction, heating of surrounding bodies, etc. The efficiency is always less than unity. Accordingly, the efficiency is expressed as a fraction of the energy expended, that is, as a proper fraction or as a percentage, and is a dimensionless quantity. Efficiency characterizes how efficiently a machine or mechanism works. The efficiency of thermal power plants reaches 35-40%, internal combustion engines with supercharging and pre-cooling - 40-50%, dynamos and high-power generators - 95%, transformers - 98%.

The task in which you need to find the efficiency or it is known, you need to start with a logical reasoning - what work is useful and what is spent.

Law of conservation of mechanical energy

full mechanical energy the sum of kinetic energy (i.e., the energy of motion) and potential (i.e., the energy of interaction of bodies by the forces of gravity and elasticity) is called:

If mechanical energy does not pass into other forms, for example, into internal (thermal) energy, then the sum of kinetic and potential energy remains unchanged. If mechanical energy is converted into thermal energy, then the change in mechanical energy is equal to the work of the friction force or energy losses, or the amount of heat released, and so on, in other words, the change in total mechanical energy is equal to the work of external forces:

The sum of the kinetic and potential energies of the bodies that make up a closed system (i.e., one in which no external forces act, and their work is equal to zero, respectively) and interacting with each other by gravitational forces and elastic forces, remains unchanged:

This statement expresses law of conservation of energy (LSE) in mechanical processes. It is a consequence of Newton's laws. The law of conservation of mechanical energy is fulfilled only when the bodies in a closed system interact with each other by forces of elasticity and gravity. In all problems on the law of conservation of energy there will always be at least two states of the system of bodies. The law says that the total energy of the first state will be equal to the total energy of the second state.

Algorithm for solving problems on the law of conservation of energy:

Find the points of the initial and final position of the body.
Write down what or what energies the body has at these points.
Equate the initial and final energy of the body.
Add other necessary equations from previous physics topics.
Solve the resulting equation or system of equations using mathematical methods.

It is important to note that the law of conservation of mechanical energy made it possible to obtain a connection between the coordinates and velocities of the body in two different points trajectories without analyzing the law of motion of the body at all intermediate points. The application of the law of conservation of mechanical energy can greatly simplify the solution of many problems.

In real conditions, almost always moving bodies, along with gravitational forces, elastic forces and other forces, are acted upon by friction forces or resistance forces of the medium. The work of the friction force depends on the length of the path.

If friction forces act between the bodies that make up a closed system, then mechanical energy is not conserved. Part of the mechanical energy is converted into internal energy bodies (heating). Thus, the energy as a whole (i.e. not only mechanical energy) is conserved in any case.

For any physical interactions energy does not arise and does not disappear. It only changes from one form to another. This experimentally established fact expresses the fundamental law of nature - law of conservation and transformation of energy.

One of the consequences of the law of conservation and transformation of energy is the statement about the impossibility of creating " perpetual motion machine» (perpetuum mobile) - a machine that could do work indefinitely without expending energy.

Miscellaneous work tasks

If you need to find mechanical work in the problem, then first select the method for finding it:

Jobs can be found using the formula: A = FS cos α . Find the force that does the work and the amount of displacement of the body under the action of this force in the selected reference frame. Note that the angle must be chosen between the force and displacement vectors.
The work of an external force can be found as the difference between the mechanical energy in the final and initial situations. Mechanical energy is equal to the sum of the kinetic and potential energies of the body.
The work done to lift a body at a constant speed can be found by the formula: A = mgh, where h- the height to which it rises center of gravity of the body.
Work can be found as the product of power and time, i.e. according to the formula: A = Pt.
Work can be found as the area of a figure under a graph of force versus displacement or power versus time.

The law of conservation of energy and the dynamics of rotational motion

The tasks of this topic are quite complex mathematically, but with knowledge of the approach they are solved according to a completely standard algorithm. In all problems you will have to consider the rotation of the body in the vertical plane. The solution will be reduced to the following sequence of actions:

It is necessary to determine the point of interest to you (the point at which it is necessary to determine the speed of the body, the force of the thread tension, weight, and so on).
Write down Newton's second law at this point, given that the body rotates, that is, it has centripetal acceleration.
Write down the law of conservation of mechanical energy so that it contains the speed of the body in the same point of interest, as well as the characteristics of the state of the body in some state about which something is known.
Depending on the condition, express the speed squared from one equation and substitute it into another.
Carry out other necessary mathematical operations to get the final result.

When solving problems, remember that:

The condition for passing the upper point during rotation on the threads at a minimum speed is the reaction force of the support N at the top point is 0. The same condition is met when passing through the top point of the dead loop.
When rotating on a rod, the condition for passing the entire circle is: the minimum speed at the top point is 0.
The condition for the separation of the body from the surface of the sphere is that the reaction force of the support at the separation point is zero.

Inelastic Collisions

The law of conservation of mechanical energy and the law of conservation of momentum make it possible to find solutions to mechanical problems in cases where the acting forces are unknown. An example of such problems is the impact interaction of bodies.

Impact (or collision) It is customary to call the short-term interaction of bodies, as a result of which their velocities experience significant changes. During the collision of bodies, short-term impact forces act between them, the magnitude of which, as a rule, is unknown. Therefore, it is impossible to consider the impact interaction directly with the help of Newton's laws. The application of the laws of conservation of energy and momentum in many cases makes it possible to exclude the process of collision from consideration and obtain a relationship between the velocities of bodies before and after the collision, bypassing all intermediate values of these quantities.

One often has to deal with the impact interaction of bodies in everyday life, in technology and in physics (especially in the physics of the atom and elementary particles). In mechanics, two models of impact interaction are often used - absolutely elastic and absolutely inelastic impacts.

Absolutely inelastic impact Such a shock interaction is called, in which the bodies are connected (stick together) with each other and move on as one body.

In a perfectly inelastic impact, mechanical energy is not conserved. It partially or completely passes into the internal energy of bodies (heating). To describe any impacts, you need to write down both the law of conservation of momentum and the law of conservation of mechanical energy, taking into account the released heat (it is highly desirable to draw a drawing beforehand).

Absolutely elastic impact

Absolutely elastic impact is called a collision in which the mechanical energy of a system of bodies is conserved. In many cases, collisions of atoms, molecules and elementary particles obey the laws of absolutely elastic impact. With an absolutely elastic impact, along with the law of conservation of momentum, the law of conservation of mechanical energy is fulfilled. A simple example An absolutely elastic collision can be the central impact of two billiard balls, one of which was at rest before the collision.

center punch balls is called a collision, in which the speeds of the balls before and after the impact are directed along the line of centers. Thus, using the laws of conservation of mechanical energy and momentum, it is possible to determine the velocities of the balls after the collision, if their velocities before the collision are known. Center punch is very rarely implemented in practice, especially if we are talking about collisions of atoms or molecules. In non-central elastic collision, the velocities of particles (balls) before and after the collision are not directed along the same straight line.

A special case of a non-central elastic impact is the collision of two billiard balls of the same mass, one of which was stationary before the collision, and the speed of the second was not directed along the line of the centers of the balls. In this case, the velocity vectors of the balls after elastic collision are always directed perpendicular to each other.

Conservation laws. Difficult tasks

Multiple bodies

In some tasks on the law of conservation of energy, the cables with the help of which certain objects move can have mass (that is, not be weightless, as you might already be used to). In this case, the work of moving such cables (namely, their centers of gravity) must also be taken into account.

If two bodies connected by a weightless rod rotate in a vertical plane, then:

choose a zero level for calculating potential energy, for example, at the level of the axis of rotation or at the level of the lowest point where one of the loads is located and make a drawing;
the law of conservation of mechanical energy is written, in which the sum of the kinetic and potential energies of both bodies in the initial situation is written on the left side, and the sum of the kinetic and potential energies of both bodies in the final situation is written on the right side;
take into account that the angular velocities of the bodies are the same, then the linear velocities of the bodies are proportional to the radii of rotation;
if necessary, write down Newton's second law for each of the bodies separately.

Projectile burst

In the event of a projectile burst, explosive energy is released. To find this energy, it is necessary to subtract the mechanical energy of the projectile before the explosion from the sum of the mechanical energies of the fragments after the explosion. We will also use the momentum conservation law written in the form of the cosine theorem (vector method) or in the form of projections onto selected axes.

Collisions with a heavy plate

Let towards a heavy plate that moves at a speed v, a light ball of mass moves m with speed u n. Since the momentum of the ball is much less than the momentum of the plate, the plate's speed will not change after impact, and it will continue to move at the same speed and in the same direction. As a result of elastic impact, the ball will fly off the plate. Here it is important to understand that the speed of the ball relative to the plate will not change. In this case, for the final speed of the ball we get:

Thus, the speed of the ball after impact is increased by twice the speed of the wall. A similar reasoning for the case when the ball and the plate were moving in the same direction before the impact leads to the result that the speed of the ball is reduced by twice the speed of the wall:

In physics and mathematics, among other things, three essential conditions must be met:

Study all the topics and complete all the tests and tasks given in the study materials on this site. To do this, you need nothing at all, namely: to devote three to four hours every day to preparing for the CT in physics and mathematics, studying theory and solving problems. The fact is that the CT is an exam where it is not enough just to know physics or mathematics, you also need to be able to quickly and without failures solve a large number of tasks for different topics and varying complexity. The latter can only be learned by solving thousands of problems.
Learn all formulas and laws in physics, and formulas and methods in mathematics. In fact, it is also very simple to do this, there are only about 200 necessary formulas in physics, and even a little less in mathematics. In each of these subjects there are about a dozen standard methods for solving problems of a basic level of complexity, which can also be learned, and thus, completely automatically and without difficulty, solve most of the digital transformation at the right time. After that, you will only have to think about the most difficult tasks.
Attend all three stages of rehearsal testing in physics and mathematics. Each RT can be visited twice to solve both options. Again, on the DT, in addition to the ability to quickly and efficiently solve problems, and the knowledge of formulas and methods, it is also necessary to be able to properly plan time, distribute forces, and most importantly fill out the answer form correctly, without confusing either the numbers of answers and tasks, or your own surname. Also, during the RT, it is important to get used to the style of posing questions in tasks, which may seem very unusual to an unprepared person on the DT.

Successful, diligent and responsible implementation of these three points will allow you to show an excellent result on the CT, the maximum of what you are capable of.

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When bodies interact pulse one body can be partially or completely transferred to another body. If external forces from other bodies do not act on a system of bodies, such a system is called closed.

This fundamental law of nature is called the law of conservation of momentum. It is a consequence of the second and third Newton's laws.

Consider any two interacting bodies that are part of a closed system. The forces of interaction between these bodies will be denoted by and According to Newton's third law If these bodies interact during time t, then the impulses of the interaction forces are identical in absolute value and directed in opposite directions: Let's apply Newton's second law to these bodies:

where and are the momenta of the bodies at the initial moment of time, and are the momenta of the bodies at the end of the interaction. From these ratios it follows:

This equality means that as a result of the interaction of two bodies, their total momentum has not changed. Considering now all possible pair interactions of bodies included in a closed system, we can conclude that the internal forces of a closed system cannot change its total momentum, that is, the vector sum of the momentums of all bodies included in this system.

Mechanical work and power

The energy characteristics of motion are introduced on the basis of the concept mechanical work or work of force.

Work A done by a constant force called a physical quantity equal to the product of the modules of force and displacement, multiplied by the cosine of the angle α between the force vectors and displacement(Fig. 1.1.9):

Work is a scalar quantity. It can be both positive (0° ≤ α< 90°), так и отрицательна (90° < α ≤ 180°). При α = 90° работа, совершаемая силой, равна нулю. В системе СИ работа измеряется в joules (J).

A joule is equal to the work done by a force of 1 N in a displacement of 1 m in the direction of the force.

If the projection of the force on the direction of movement does not remain constant, the work should be calculated for small displacements and summarize the results:

An example of a force whose modulus depends on the coordinate is the elastic force of a spring obeying Hooke's law. In order to stretch the spring, an external force must be applied to it, the modulus of which is proportional to the elongation of the spring (Fig. 1.1.11).

The dependence of the module of the external force on the x coordinate is shown on the graph by a straight line (Fig. 1.1.12).

According to the area of the triangle in Fig. 1.18.4, you can determine the work done by an external force applied to the right free end of the spring:

The same formula expresses the work done by an external force when the spring is compressed. In both cases, the work of the elastic force is equal in absolute value to the work of the external force and opposite in sign.

If several forces are applied to the body, then general work of all forces is equal to the algebraic sum of the work performed by individual forces, and is equal to the work resultant of applied forces.

The work done by a force per unit of time is called power. Power N is a physical quantity equal to the ratio of work A to the time interval t during which this work is done.