Note that work and energy have the same units of measurement. This means that work can be converted into energy. For example, in order to raise a body to a certain height, then it will have potential energy, a force is needed that will do this work. The work done by the lifting force will turn into potential energy.
The rule for determining work according to the dependence graph F(r): the work is numerically equal to the area of the figure under the graph of force versus displacement.
Angle between force vector and displacement
1) Correctly determine the direction of the force that does the work; 2) We depict the displacement vector; 3) We transfer the vectors to one point and obtain the desired angle.
In the figure, the body is acted upon by the force of gravity (mg), the reaction of the support (N), the force of friction (Ftr) and the tension force of the rope F, under the influence of which the body moves r.
Work of gravity
Ground reaction work
Work of friction force
Work done by rope tension
Work done by resultant force
The work done by the resultant force can be found in two ways: 1st method - as the sum of the work (taking into account the “+” or “-” signs) of all forces acting on the body, in our example
Method 2 - first of all, find the resultant force, then directly its work, see figure
Work of elastic force
To find the work done by the elastic force, it is necessary to take into account that this force changes because it depends on the elongation of the spring. From Hooke's law it follows that as the absolute elongation increases, the force increases.
To calculate the work of the elastic force during the transition of a spring (body) from an undeformed state to a deformed state, use the formula
Power
A scalar quantity that characterizes the speed of work (an analogy can be drawn with acceleration, which characterizes the rate of change in speed). Determined by the formula
Efficiency
Efficiency is the ratio of the useful work done by a machine to all the work expended (energy supplied) during the same time
The efficiency is expressed as a percentage. The closer this number is to 100%, the higher the machine's performance. There cannot be an efficiency greater than 100, since it is impossible to do more work using less energy.
The efficiency of an inclined plane is the ratio of the work done by gravity to the work expended in moving along the inclined plane.
The main thing to remember
1) Formulas and units of measurement;
2) The work is performed by force;
3) Be able to determine the angle between the force and displacement vectors
If the work done by a force when moving a body along a closed path is zero, then such forces are called conservative or potential. The work done by the friction force when moving a body along a closed path is never equal to zero. The friction force, unlike the force of gravity or elastic force, is non-conservative or non-potential.
There are conditions under which the formula cannot be used 
If the force is variable, if the trajectory of movement is a curved line. In this case, the path is divided into small sections for which these conditions are met, and the elementary work on each of these sections is calculated. Full work in this case is equal to the algebraic sum of elementary works: 
The value of the work done by a certain force depends on the choice of reference system.
Content:Electric current is generated in order to be used in the future for certain purposes, to perform some kind of work. Thanks to electricity, all devices, devices and equipment function. The work itself represents a certain effort made to move electric charge on set distance. Conventionally, such work within a section of the circuit will be equal to the numerical value of the voltage in this section.
To perform the necessary calculations, you need to know how the work of the current is measured. All calculations are carried out based on initial data obtained using measuring instruments. The larger the charge, the more effort is required to move it, the more big job will be completed.
What is the work of current called?
Electric current is like physical quantity, in itself has no practical significance. The most important factor is the effect of the current, characterized by the work it performs. The work itself represents certain actions during which one type of energy is transformed into another. For example, electrical energy is converted into mechanical energy by rotating the motor shaft. The work itself electric current consists in the movement of charges in a conductor under the influence electric field. In fact, all the work of moving charged particles is done by the electric field.
In order to perform calculations, a formula for the operation of electric current must be derived. To compile formulas, you will need parameters such as current strength and. Since the work done by an electric current and the work done by an electric field are the same thing, it will be expressed as the product of the voltage and the charge flowing in the conductor. That is: A = Uq. This formula was derived from the relationship that determines the voltage in the conductor: U = A/q. It follows that voltage represents the work done by the electric field A to transport a charged particle q.
The charged particle or charge itself is displayed as the product of the current strength and the time spent on the movement of this charge along the conductor: q = It. In this formula, the relation for the current strength in the conductor was used: I = q/t. That is, it is the ratio of the charge to the period of time during which the charge passes through cross section conductor. In its final form, the formula for the work of electric current will look like the product of known quantities: A = UIt.
In what units is the work of electric current measured?
Before directly addressing the question of how the work of electric current is measured, it is necessary to collect the units of measurement of all physical quantities with which this parameter is calculated. Any work, therefore, the unit of measurement of this quantity will be 1 Joule (1 J). Voltage is measured in volts, current is measured in amperes, and time is measured in seconds. This means the unit of measurement will look like this: 1 J = 1V x 1A x 1s.
Based on the obtained units of measurement, the work of electric current will be determined as the product of the current strength in a section of the circuit, the voltage at the ends of the section and the period of time during which the current flows through the conductor.
Measurements are carried out using a voltmeter and a clock. These devices allow you to effectively solve the problem of how to find exact value this parameter. When connecting an ammeter and a voltmeter to a circuit, it is necessary to monitor their readings for a specified period of time. The obtained data is inserted into the formula, after which the final result is displayed.
The functions of all three devices are combined in electric meters that take into account the energy consumed, and in fact the work done by electric current. Here another unit is used - 1 kW x h, which also means how much work was done during a unit of time.
If a force acts on a body, then this force does work to move the body. Before defining work during curvilinear motion of a material point, let us 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.
and work of force 
by moving a material point from a point 1
exactly 2
defined as a curvilinear integral:
–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, you can increase the energy reserve in any device. In other words, work is a quantitative measure of energy change:
.
Energy, like work, is measured in SI 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. Consider a material point moving under the influence of a 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:
–total mechanical energy of the 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 there are no other external forces, then 
, i.e. in this case, the total mechanical energy of the body is conserved.
1.5. MECHANICAL WORK AND KINETIC ENERGY
The concept of energy. Mechanical energy. Work is a quantitative measure of energy change. Work of resultant forces. Work of forces in mechanics. The concept of power. Kinetic energy as a measure of mechanical motion. Communication change ki netic energy with the work of internal and external forces.Kinetic energy of a system in various reference systems.Koenig's theorem.
Energy - it is a universal measure of various forms of movement and interaction. M mechanical energy describes the amount potentialAndkinetic energy, available in the components mechanical system . Mechanical energy- this is the energy associated with the movement of an object or its position, the ability to perform mechanical work.
Work of force - this is a quantitative characteristic of the process of energy exchange between interacting bodies.
Let a particle, under the influence of a force, move along a certain trajectory 1-2 (Fig. 5.1). IN general case power in the process
The movement of a particle can change both in magnitude and in direction. Let us consider, as shown in Fig. 5.1, an elementary displacement within which the force can be considered constant.
The effect of force on displacement is characterized by a value equal to the scalar product, which is called basic work moving forces. It can be presented in another form:
,
where is the angle between the vectors and is the elementary path, the projection of the vector onto the vector is indicated (Fig. 5.1).
So, the elementary work of force on displacement
|
|
.
The quantity is algebraic: depending on the angle between the force vectors and or on the sign of the projection of the force vector onto the displacement vector, it can be either positive or negative and, in particular, equal to zero if i.e. . The SI unit of work is the Joule, abbreviated J.
By summing (integrating) expression (5.1) over all elementary sections of the path from point 1 to point 2, we find the work done by the force on a given displacement:
it is clear that the elementary work A is numerically equal to the area of the shaded strip, and the work A on the path from point 1 to point 2 is the area of the figure bounded by the curve, ordinates 1 and 2 and the s axis. In this case, the area of the figure above the s-axis is taken with a plus sign (it corresponds to positive work), and the area of the figure under the s-axis is taken with a minus sign (it corresponds to negative work).
Let's look at examples of how to calculate work. Work of elastic force where is the radius vector of particle A relative to point O (Fig. 5.3).
Let us move particle A, which is acted upon by this force, along an arbitrary path from point 1 to point 2. Let us first find the elementary work of force on elementary displacement:
.
Scalar product 
where is the projection of the displacement vector onto the vector . This projection is equal to the increment of the modulus of the vector. Therefore,
Now let’s calculate the work done by this force along the entire path, i.e., integrate the last expression from point 1 to point 2:
|
|
Let's calculate the work done by the gravitational (or the mathematically analogous Coulomb force) force. Let there be a stationary point mass (point charge) at the beginning of the vector (Fig. 5.3). Let us determine the work done by the gravitational (Coulomb) force when particle A moves from point 1 to point 2 along an arbitrary path. The force acting on particle A can be represented as follows:
where the parameter for the gravitational interaction is equal to , and for the Coulomb interaction its value is equal to . Let us first calculate the elementary work of this force on displacement
As in the previous case, the scalar product is therefore
.
The work of this force all the way from point 1 to point 2
|
|
Let us now consider the work of a uniform force of gravity. Let us write this force in the form where unit vertical axis z with a positive direction is indicated (Fig. 5.4). Elementary work of gravity on displacement
Scalar product 
where the projection onto the unit unit is equal to the increment of the z coordinate. Therefore, the expression for work takes the form
The work done by a given force all the way from point 1 to point 2
|
|
The forces considered are interesting in the sense that their work, as can be seen from formulas (5.3) - (5.5), does not depend on the shape of the path between points 1 and 2, but depends only on the position of these points. This very important feature of these forces is not inherent, however, to all forces. For example, the friction force does not have this property: the work of this force depends not only on the position of the starting and ending points, but also on the shape of the path between them.
Until now we have been talking about the work of one force. If several forces act on a particle in the process of motion, the resultant of which is then it is easy to show that the work of the resulting force on a certain displacement is equal to the algebraic sum of the work performed by each of the forces separately on the same displacement. Really,
Let us introduce a new quantity into consideration - power. It is used to characterize the speed at which work is done. Power , a-priory, - is the work done by a force per unit time . If a force does work over a period of time, then the power developed by this force in this moment time, there is Considering that , we get
The SI unit of power is Watt, abbreviated as W.
Thus, the power developed by force is equal to the scalar product of the force vector and the speed vector with which the point of application of this force moves. Like work, power is an algebraic quantity.
Knowing the power of the force, you can find the work done by this force over a period of time t. Indeed, presenting the integrand in (5.2) as 
we get
You should also pay attention to one very significant circumstance. When talking about work (or power), it is necessary in each specific case to clearly indicate or imagine the work what kind of strength(or forces) is meant. Otherwise, as a rule, misunderstandings are inevitable.
Let's consider the concept particle kinetic energy. Let a particle of mass T moves under the influence of some force (in the general case, this force can be the result of several forces). Let's find the elementary work that this force does on an elementary displacement. Keeping in mind that and , we write
.
Scalar product 
where is the projection of the vector onto the direction of the vector. This projection is equal to the increment of the magnitude of the velocity vector. Therefore, the elementary work
From this it is clear that the work of the resulting force goes to increase a certain value in parentheses, which is called kinetic energy particles.
and upon final movement from point 1 to point 2
|
|
(5. 10 ) |
i.e. the increment in the kinetic energy of a particle at a certain displacement is equal to the algebraic sum of the work of all forces, acting on the particle at the same displacement. If then, that is, the kinetic energy of the particle increases; if that is, then the kinetic energy decreases.
Equation (5.9) can be presented in another form by dividing both sides by the corresponding time interval dt:
|
|
(5. 11 ) |
This means that the derivative of the kinetic energy of a particle with respect to time is equal to the power N of the resulting force acting on the particle.
Now let's introduce the concept kinetic energy of the system . Let us consider an arbitrary system of particles in a certain reference frame. Let a particle of the system have kinetic energy at a given moment. The increment in the kinetic energy of each particle is equal, according to (5.9), to the work of all forces acting on this particle: Let us find the elementary work performed by all forces acting on all particles of the system:
where is the total kinetic energy of the system. Note that the kinetic energy of the system is the quantity additive : it is equal to the sum of the kinetic energies of the individual parts of the system, regardless of whether they interact with each other or not.
So, the increase in the kinetic energy of the system is equal to the work done by all the forces acting on all particles of the system. With the elementary movement of all particles
|
(5.1 2 ) |
and at the final movement
i.e. the time derivative of the kinetic energy of the system is equal to the total power of all forces acting on all particles of the system,
Koenig's theorem: kinetic energy K systems of particles can be represented as the sum of two terms: a) kinetic energy mV c 2 /2 an imaginary material point whose mass is equal to the mass of the entire system, and whose speed coincides with the speed of the center of mass; b) kinetic energy K rel system of particles calculated in the center of mass system.
Basic theoretical information
Mechanical work
The energy characteristics of motion are introduced based on the concept mechanical work or force work. Work done by a constant force F, is a physical quantity equal to the product of the force and displacement moduli multiplied by the cosine of the angle between the force vectors F and movements S:
Work is a scalar quantity. 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). Joule equal to work performed by a force of 1 newton for a displacement of 1 meter in the direction of the force.
If the force changes over time, then to find the work, build a graph of the force versus displacement and find the area of the 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 control = kx).
Power
The work done by a force per unit time is called power. Power P(sometimes denoted by the letter N) – physical quantity equal to the work ratio A to a period of time 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(if, 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:
Using 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 do you know what power to count? If the problem asks for power at a moment in time or at some point in space, then instantaneous is considered. If they ask about power over a certain period of time or part of the route, then look for average power.
Efficiency - efficiency factor, is equal to the ratio of useful work to expended, or useful power to expended:
Which work is useful and which is wasted is determined from the conditions of a specific task through logical reasoning. For example, if crane does work to lift the load to a certain height, then the work to lift the load will be useful (since it is for this purpose that the crane was created), and the work done by the electric motor of the crane 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 purpose of doing the work ( useful work or power), and what was the mechanism or way of doing all the work (power expended or work).
In general, efficiency shows how efficiently a 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 a body’s mass and the square of its speed is called kinetic energy of the body (energy of movement):
That is, if a car weighing 2000 kg moves at a speed of 10 m/s, then it has kinetic energy equal to E k = 100 kJ and is capable of doing 100 kJ of work. This energy can turn into heat (when a car brakes, the tires of the wheels, the road and the brake discs heat up) or can be spent on deforming the car and the body that the car collides 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. 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 a mass m began to move at speed v it is necessary to do work equal to the obtained value of kinetic energy. If the body has a mass m moves at speed v, then to stop it it is necessary to do work equal to its initial kinetic energy. When braking, kinetic energy is mainly (except for cases of impact, when the energy goes to deformation) “taken away” by the friction force.
Theorem on kinetic energy: the work of the resultant force is equal to the change in the kinetic energy of the body:
The theorem on kinetic energy is also valid in the general case, when a body moves under the influence 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 involving acceleration and deceleration of a body.
Potential energy
Along with kinetic energy or energy of motion in physics important role plays concept potential energy or energy of interaction of bodies.
Potential energy is determined by the relative position of bodies (for example, the position of the body relative to the surface of the Earth). 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 done by such forces on a closed trajectory is zero. This property is possessed by gravity and elastic force. 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– 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 to zero level. The work done by gravity is equal to the change in the potential energy of the body, taken with the opposite sign:
Often in energy problems one has to find the work of lifting (turning over, getting out of a hole) 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. What has a physical meaning is not the potential energy itself, but its change when a body moves from one position to another. This change is independent of the choice of zero level.
Potential energy of a stretched spring calculated by the formula:
Where: k– spring stiffness. An extended (or compressed) spring can set a body attached to it in motion, that is, impart kinetic energy to this body. Consequently, such a spring has a reserve of energy. Tension 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 done by 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 at elastic deformation– this 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 path traveled (this type of force, whose work depends on the trajectory and the path traveled is called: dissipative forces). The concept of potential energy for the friction force cannot be introduced.
Efficiency
Efficiency factor (efficiency)– characteristic of the efficiency of a system (device, machine) in relation to the conversion or transmission of energy. It is determined by the ratio of usefully used energy to the total amount of energy received by the system (the formula has already been given above).
Efficiency can be calculated both through work and through power. Useful and expended work (power) are always determined by simple logical reasoning.
In electric motors, efficiency is 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– the ratio of 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 such various systems, such as nuclear reactors, electric generators and engines, thermal power plants, semiconductor devices, biological objects, etc.
Due to inevitable energy losses due to friction, heating of surrounding bodies, etc. Efficiency is always less than unity. Accordingly, 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 operates. 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%.
A problem in which you need to find the efficiency or it is known, you need to start with logical reasoning - which work is useful and which is wasted.
Law of conservation of mechanical energy
Total mechanical energy is called 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):
If mechanical energy does not transform into other forms, for example, into internal (thermal) energy, then the sum of kinetic and potential energy remains unchanged. If mechanical energy turns 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 energy of the bodies that make up closed system(i.e. one in which there are no external forces acting, and their work is correspondingly equal to zero) and the gravitational and elastic forces interacting with each other, remains unchanged:
This statement expresses law of conservation of energy (LEC) in mechanical processes. It is a consequence of Newton's laws. The law of conservation of mechanical energy is satisfied only when 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 a system of bodies. The law states 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 relationship between the coordinates and velocities of a body at two different points of the trajectory 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 frictional forces or environmental resistance forces. The work done by 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, energy as a whole (i.e., not only mechanical) is conserved in any case.
During any physical interactions, energy neither appears nor disappears. It just changes from one form to another. This experimentally established fact expresses a 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 perform work indefinitely without consuming energy.
Various tasks for work
If the problem requires finding mechanical work, then first select a method for finding it:
- A job 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 influence of this force in the chosen frame of reference. Note that the angle must be chosen between the force and displacement vectors.
- The work done by an external force can be found as the difference in 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 using the formula: A = mgh, Where h- height to which it rises body center of gravity.
- Work can be found as the product of power and time, i.e. according to the formula: A = Pt.
- The work can be found as the area of the figure under the graph of force versus displacement or power versus time.
Law of conservation of energy and dynamics of rotational motion
The problems of this topic are quite complex mathematically, but if you know the approach, they can be solved using a completely standard algorithm. In all problems you will have to consider the rotation of the body in the vertical plane. The solution will come down to the following sequence of actions:
- You need to determine the point you are interested in (the point at which you need to determine the speed of the body, the tension force of the thread, weight, and so on).
- Write down Newton’s second law at this point, taking into account 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 at that very interesting point, 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 squared speed from one equation and substitute it into the other.
- Carry out the remaining necessary mathematical operations to obtain the final result.
When solving problems, you need to remember that:
- The condition for passing the top point when rotating on a thread at a minimum speed is the support reaction force N at the top point is 0. The same condition is met when passing the top dead center loops.
- 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 a body from the surface of the sphere is that the support reaction force 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 this type of problem is the impact interaction of bodies.
By impact (or collision) It is customary to call a short-term interaction of bodies, as a result of which their speeds experience significant changes. During a collision between bodies, short-term strike forces, the magnitude of which is usually unknown. Therefore, it is impossible to consider the impact interaction directly using Newton's laws. The application of the laws of conservation of energy and momentum in many cases makes it possible to exclude the collision process itself from consideration and obtain a connection between the velocities of bodies before and after the collision, bypassing all intermediate values of these quantities.
We often have 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 They call this impact interaction in which bodies connect (stick together) with each other and move on as one body.
In a completely inelastic collision, mechanical energy is not conserved. It partially or completely turns 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 heat released (it is highly advisable to make a drawing first).
Absolutely elastic impact
Absolutely elastic impact 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 satisfied. A simple example A perfectly elastic collision can be a central impact of two billiard balls, one of which was at rest before the collision.
Central strike balls is called a collision in which the velocities 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 a collision if their velocities before the collision are known. The central strike is very rarely implemented in practice, especially if we're talking about about collisions of atoms or molecules. In a non-central elastic collision, the velocities of particles (balls) before and after the collision are not directed in one straight line.
A special case of an off-central elastic impact can be the collision of two billiard balls of the same mass, one of which was motionless 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 an elastic collision are always directed perpendicular to each other.
Conservation laws. Complex tasks
Multiple bodies
In some problems on the law of conservation of energy, the cables with which certain objects are moved 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) also needs to be taken into account.
If two bodies connected by a weightless rod rotate in a vertical plane, then:
- choose a zero level to calculate potential energy, for example at the level of the axis of rotation or at the level of the lowest point of one of the weights and be sure to make a drawing;
- write down the law of conservation of mechanical energy, in which on the left side we write the sum of the kinetic and potential energy of both bodies in the initial situation, and on the right side we write the sum of the kinetic and potential energy of both bodies in the final situation;
- take into account that angular velocities bodies are identical, 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.
Shell burst
When a projectile explodes, 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 law of conservation of momentum, 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 us meet a heavy plate that moves at 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, after the impact the speed of the plate will not change, and it will continue to move at the same speed and in the same direction. As a result of the elastic impact, the ball will fly away from the plate. It is important to understand here that the speed of the ball relative to the plate will not change. In this case, for the final speed of the ball we obtain:
Thus, the speed of the ball after impact increases by twice the speed of the wall. Similar reasoning for the case when before the impact the ball and the plate were moving in the same direction leads to the result that the speed of the ball decreases by twice the speed of the wall:
In physics and mathematics, among other things, three most important conditions must be met:
- Study all topics and complete all tests and assignments given in the educational materials on this site. To do this, you need nothing at all, namely: 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 CT is an exam where it is not enough just to know physics or mathematics, you also need to be able to solve it quickly and without failures a large number of tasks for different topics and of varying complexity. The latter can only be learned by solving thousands of problems.
- Learn all the formulas and laws in physics, and formulas and methods in mathematics. In fact, this is also very simple to do; 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 solving most of the CT at the right time. After this, 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 decide on both options. Again, on the CT, in addition to the ability to quickly and efficiently solve problems, and knowledge of formulas and methods, you must also be able to properly plan time, distribute forces, and most importantly, correctly fill out the answer form, without confusing the numbers of answers and problems, or your own last name. Also, during RT, it is important to get used to the style of asking questions in problems, which may seem very unusual to an unprepared person at the DT.
Successful, diligent and responsible implementation of these three points will allow you to show an excellent result at the CT, the maximum of what you are capable of.
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