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Two cases of transformation of the mechanical motion of a material point or system of points:
- mechanical motion is transferred from one mechanical system to another as mechanical motion;
- mechanical motion turns into another form of motion of matter (into the form of potential energy, heat, electricity, etc.).
When the transformation of mechanical motion without its transition to another form of motion is considered, the measure of mechanical motion is the vector of momentum of a material point or mechanical system. The measure of the force in this case is the vector of the force impulse.
When mechanical motion turns into another form of motion of matter, the kinetic energy of a material point or mechanical system acts as a measure of mechanical motion. The measure of the action of force when transforming mechanical motion into another form of motion is the work of force
Kinetic energy
Kinetic energy is the body's ability to overcome an obstacle while moving.
Kinetic energy of a material point
The kinetic energy of a material point is a scalar quantity that is equal to half the product of the mass of the point and the square of its speed.
Kinetic energy:
- characterizes both translational and rotational movements;
- does not depend on the direction of movement of the points of the system and does not characterize changes in these directions;
- characterizes the action of both internal and external forces.
Kinetic energy of a mechanical system
The kinetic energy of the system is equal to the sum of the kinetic energies of the bodies of the system. Kinetic energy depends on the type of motion of the bodies of the system.
Determination of the kinetic energy of a solid body for different types of motion.
Kinetic energy of translational motion
During translational motion, the kinetic energy of the body is equal to T=m V 2 /2.
The measure of the inertia of a body during translational motion is mass.
Kinetic energy of rotational motion of a body
During the rotational motion of a body, kinetic energy is equal to half the product of the moment of inertia of the body relative to the axis of rotation and the square of its angular velocity.
A measure of the inertia of a body during rotational motion is the moment of inertia.
The kinetic energy of a body does not depend on the direction of rotation of the body.
Kinetic energy of plane-parallel motion of a body
With plane-parallel motion of a body, the kinetic energy is equal to
Work of force
The work of force characterizes the action of a force on a body during some movement and determines the change in the velocity modulus of a moving point.
Elementary work of force
The elementary work of a force is defined as a scalar quantity equal to the product of the projection of the force onto the tangent to the trajectory, directed in the direction of motion of the point, and the infinitesimal displacement of the point, directed along this tangent.
Work done by force on final displacement
The work done by a force on a final displacement is equal to the sum of its work on elementary sections.
The work of a force on a final displacement M 1 M 0 is equal to the integral of the elementary work along this displacement.
The work of a force on displacement M 1 M 2 is depicted by the area of the figure, limited by the abscissa axis, the curve and the ordinates corresponding to the points M 1 and M 0.
The unit of measurement for the work of force and kinetic energy in the SI system is 1 (J).
Theorems about the work of force
Theorem 1. The work done by the resultant force on a certain displacement is equal to the algebraic sum of the work done by the component forces on the same displacement.
Theorem 2. The work done by a constant force on the resulting displacement is equal to the algebraic sum of the work done by this force on the component displacements.
Power
Power is a quantity that determines the work done by a force per unit of time.
The unit of power measurement is 1W = 1 J/s.
Cases of determining the work of forces
Work of internal forces
The sum of the work done by the internal forces of a rigid body during any movement is zero.
Work of gravity
Work of elastic force
Work of friction force
Work of forces applied to a rotating body
The elementary work of forces applied to a rigid body rotating around a fixed axis is equal to the product of the main moment of external forces relative to the axis of rotation and the increment in the angle of rotation.
Rolling resistance
In the contact zone of the stationary cylinder and the plane, local deformation of contact compression occurs, the stress is distributed according to an elliptical law, and the line of action of the resultant N of these stresses coincides with the line of action of the load force on the cylinder Q. When the cylinder rolls, the load distribution becomes asymmetrical with a maximum shifted towards movement. The resultant N is displaced by the amount k - the arm of the rolling friction force, which is also called the rolling friction coefficient and has the dimension of length (cm)
Theorem on the change in kinetic energy of a material point
The change in the kinetic energy of a material point at a certain displacement is equal to the algebraic sum of all forces acting on the point at the same displacement.
Theorem on the change in kinetic energy of a mechanical system
The change in the kinetic energy of a mechanical system at a certain displacement is equal to the algebraic sum of the internal and external forces acting on the material points of the system at the same displacement.
Theorem on the change in kinetic energy of a solid body
The change in the kinetic energy of a rigid body (unchanged system) at a certain displacement is equal to the sum of the external forces acting on points of the system at the same displacement.
Efficiency
Forces acting in mechanisms
Forces and pairs of forces (moments) that are applied to a mechanism or machine can be divided into groups:
1. Driving forces and moments that perform positive work (applied to the driving links, for example, gas pressure on the piston in an internal combustion engine).
2. Forces and moments of resistance that perform negative work:
- useful resistance (they perform the work required from the machine and are applied to the driven links, for example, the resistance of the load lifted by the machine),
- resistance forces (for example, friction forces, air resistance, etc.).
3. Gravity forces and elastic forces of springs (both positive and negative work, while the work for a full cycle is zero).
4. Forces and moments applied to the body or stand from the outside (reaction of the foundation, etc.), which do not do work.
5. Interaction forces between links acting in kinematic pairs.
6. The inertial forces of the links, caused by the mass and movement of the links with acceleration, can perform positive, negative work and do not perform work.
Work of forces in mechanisms
When the machine operates at a steady state, its kinetic energy does not change and the sum of the work of the driving forces and resistance forces applied to it is zero.
The work expended in setting the machine in motion is expended in overcoming useful and harmful resistances.
Mechanism efficiency
The mechanical efficiency during steady motion is equal to the ratio of the useful work of the machine to the work expended on setting the machine in motion:
Machine elements can be connected in series, parallel and mixed.
Efficiency in series connection
When mechanisms are connected in series, the overall efficiency is less than the lowest efficiency of an individual mechanism.
Efficiency in parallel connection
When mechanisms are connected in parallel, the overall efficiency is greater than the lowest and less than the highest efficiency of an individual mechanism.
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Calculation example of a spur gear
An example of calculating a spur gear. The choice of material, calculation of permissible stresses, calculation of contact and bending strength have been carried out.
An example of solving a beam bending problem
In the example, diagrams of transverse forces and bending moments were constructed, a dangerous section was found and an I-beam was selected. The problem analyzed the construction of diagrams using differential dependencies and carried out a comparative analysis of various cross sections of the beam.
An example of solving a shaft torsion problem
The task is to test the strength of a steel shaft at a given diameter, material and allowable stress. During the solution, diagrams of torques, shear stresses and twist angles are constructed. The shaft's own weight is not taken into account
An example of solving a problem of tension-compression of a rod
The task is to test the strength of a steel bar at specified permissible stresses. During the solution, diagrams of longitudinal forces, normal stresses and displacements are constructed. The rod's own weight is not taken into account
Application of the theorem on conservation of kinetic energy
An example of solving a problem using the theorem on the conservation of kinetic energy of a mechanical system
Energy is a scalar physical quantity that is a unified measure of various forms of motion of matter and a measure of the transition of the motion of matter from one form to another.
To characterize various forms of motion of matter, the corresponding types of energy are introduced, for example: mechanical, internal, energy of electrostatic, intranuclear interactions, etc.
Energy obeys the law of conservation, which is one of the most important laws of nature.
Mechanical energy E characterizes the movement and interaction of bodies and is a function of the speeds and relative positions of bodies. It is equal to the sum of kinetic and potential energies.
Kinetic energy
Let us consider the case when a body of mass m there is a constant force \(~\vec F\) (it can be the resultant of several forces) and the vectors of force \(~\vec F\) and displacement \(~\vec s\) are directed along one straight line in one direction. In this case, the work done by the force can be defined as A = F∙s. The modulus of force according to Newton's second law is equal to F = m∙a, and the displacement module s in uniformly accelerated rectilinear motion is associated with the modules of the initial υ 1 and final υ 2 speeds and accelerations A expression \(~s = \frac(\upsilon^2_2 - \upsilon^2_1)(2a)\) .
From here we get to work
\(~A = F \cdot s = m \cdot a \cdot \frac(\upsilon^2_2 - \upsilon^2_1)(2a) = \frac(m \cdot \upsilon^2_2)(2) - \frac (m \cdot \upsilon^2_1)(2)\) . (1)
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.
Kinetic energy is represented by the letter E k.
\(~E_k = \frac(m \cdot \upsilon^2)(2)\) . (2)
Then equality (1) can be written as follows:
\(~A = E_(k2) - E_(k1)\) . (3)
Kinetic energy theorem
the work of the resultant forces applied to the body is equal to the change in the kinetic energy of the body.
Since the change in kinetic energy is equal to the work done by the force (3), the kinetic energy of the body is expressed in the same units as the work, i.e., in joules.
If the initial speed of movement of a body of mass m is zero and the body increases its speed to the value υ , then the work done by the force is equal to the final value of the kinetic energy of the body:
\(~A = E_(k2) - E_(k1)= \frac(m \cdot \upsilon^2)(2) - 0 = \frac(m \cdot \upsilon^2)(2)\) . (4)
Physical meaning of kinetic energy
The kinetic energy of a body moving with a speed v shows how much work must be done by a force acting on a body at rest in order to impart this speed to it.
Potential energy
Potential energy is the energy of interaction between bodies.
The potential energy of a body raised above the Earth is the energy of interaction between the body and the Earth by gravitational forces. The potential energy of an elastically deformed body is the energy of interaction of individual parts of the body with each other by elastic forces.
Potential are called strength, the work of which depends only on the initial and final position of a moving material point or body and does not depend on the shape of the trajectory.
In a closed trajectory, the work done by the potential force is always zero. Potential forces include gravitational forces, elastic forces, electrostatic forces and some others.
Powers, the work of which depends on the shape of the trajectory, are called non-potential. When a material point or body moves along a closed trajectory, the work done by the nonpotential force is not equal to zero.
Potential energy of interaction of a body with the Earth
Let's find the work done by gravity F t when moving a body of mass m vertically down from a height h 1 above the Earth's surface to a height h 2 (Fig. 1). If the difference h 1 – h 2 is negligible compared to the distance to the center of the Earth, then the force of gravity F t during body movement can be considered constant and equal mg.
Since the displacement coincides in direction with the gravity vector, the work done by gravity is equal to
\(~A = F \cdot s = m \cdot g \cdot (h_1 - h_2)\) . (5)
Let us now consider the movement of a body along an inclined plane. When moving a body down an inclined plane (Fig. 2), the force of gravity F t = m∙g does work
\(~A = m \cdot g \cdot s \cdot \cos \alpha = m \cdot g \cdot h\) , (6)
Where h– height of the inclined plane, s– displacement module equal to the length of the inclined plane.
Movement of a body from a point IN exactly WITH along any trajectory (Fig. 3) can be mentally imagined as consisting of movements along sections of inclined planes with different heights h’, h'' etc. Work A gravity all the way from IN V WITH equal to the sum of work on individual sections of the route:
\(~A = m \cdot g \cdot h" + m \cdot g \cdot h"" + \ldots + m \cdot g \cdot h^n = m \cdot g \cdot (h" + h"" + \ldots + h^n) = m \cdot g \cdot (h_1 - h_2)\), (7)
Where h 1 and h 2 – heights from the Earth’s surface at which the points are located, respectively IN And WITH.
Equality (7) shows that the work of gravity does not depend on the trajectory of the body and is always equal to the product of the gravity modulus and the difference in heights in the initial and final positions.
When moving downward, the work of gravity is positive, when moving up it is negative. The work done by gravity on a closed trajectory is zero.
Equality (7) can be represented as follows:
\(~A = - (m \cdot g \cdot h_2 - m \cdot g \cdot h_1)\) . (8)
A physical quantity equal to the product of the mass of a body by the acceleration modulus of free fall and the height to which the body is raised above the surface of the Earth is called potential energy interaction between the body and the Earth.
Work done by gravity when moving a body of mass m from a point located at a height h 2, to a point located at a height h 1 from the Earth's surface, along any trajectory, is equal to the change in the potential energy of interaction between the body and the Earth, taken with the opposite sign.
\(~A = - (E_(p2) - E_(p1))\) . (9)
Potential energy is indicated by the letter E p.
The value of the potential energy of a body raised above the Earth depends on the choice of the zero level, i.e., the height at which the potential energy is assumed to be zero. It is usually assumed that the potential energy of a body on the Earth's surface is zero.
With this choice of the zero level, the potential energy E p of a body located at a height h above the Earth's surface, equal to the product of the mass m of the body by the absolute acceleration of free fall g and distance h it from the surface of the Earth:
\(~E_p = m \cdot g \cdot h\) . (10)
The physical meaning of the potential energy of interaction of a body with the Earth
the potential energy of a body on which gravity acts is equal to the work done by gravity when moving the body to the zero level.
Unlike the kinetic energy of translational motion, which can only have positive values, the potential energy of a body can be both positive and negative. Body mass m, located at a height h, Where h < h 0 (h 0 – zero height), has negative potential energy:
\(~E_p = -m \cdot g \cdot h\) .
Potential energy of gravitational interaction
Potential energy of gravitational interaction of a system of two material points with masses m And M, located at a distance r one from the other is equal
\(~E_p = G \cdot \frac(M \cdot m)(r)\) . (eleven)
Where G is the gravitational constant, and the zero of the potential energy reference ( E p = 0) accepted at r = ∞.
Potential energy of gravitational interaction of a body with mass m with the Earth, where h– height of the body above the Earth’s surface, M e – mass of the Earth, R e is the radius of the Earth, and the zero of the potential energy reading is chosen at h = 0.
\(~E_e = G \cdot \frac(M_e \cdot m \cdot h)(R_e \cdot (R_e +h))\) . (12)
Under the same condition of choosing zero reference, the potential energy of gravitational interaction of a body with mass m with Earth for low altitudes h (h « R e) equal
\(~E_p = m \cdot g \cdot h\) ,
where \(~g = G \cdot \frac(M_e)(R^2_e)\) is the module of gravity acceleration near the Earth's surface.
Potential energy of an elastically deformed body
Let us calculate the work done by the elastic force when the deformation (elongation) of the spring changes from some initial value x 1 to final value x 2 (Fig. 4, b, c).
The elastic force changes as the spring deforms. To find the work done by the elastic force, you can take the average value of the force modulus (since the elastic force depends linearly on x) and multiply by the displacement module:
\(~A = F_(upr-cp) \cdot (x_1 - x_2)\) , (13)
where \(~F_(upr-cp) = k \cdot \frac(x_1 - x_2)(2)\) . From here
\(~A = k \cdot \frac(x_1 - x_2)(2) \cdot (x_1 - x_2) = k \cdot \frac(x^2_1 - x^2_2)(2)\) or \(~A = -\left(\frac(k \cdot x^2_2)(2) - \frac(k \cdot x^2_1)(2) \right)\) . (14)
A physical quantity equal to half the product of the rigidity of a body by the square of its deformation is called potential energy elastically deformed body:
\(~E_p = \frac(k \cdot x^2)(2)\) . (15)
From formulas (14) and (15) it follows that the work of the elastic force is equal to the change in the potential energy of an elastically deformed body, taken with the opposite sign:
\(~A = -(E_(p2) - E_(p1))\) . (16)
If x 2 = 0 and x 1 = X, then, as can be seen from formulas (14) and (15),
\(~E_p = A\) .
Physical meaning of the potential energy of a deformed body
the potential energy of an elastically deformed body is equal to the work done by the elastic force when the body transitions to a state in which the deformation is zero.
Potential energy characterizes interacting bodies, and kinetic energy characterizes moving bodies. Both potential and kinetic energy change only as a result of such interaction of bodies in which the forces acting on the bodies do work other than zero. Let us consider the question of energy changes during the interactions of bodies forming a closed system.
Closed system- this is a system that is not acted upon by external forces or the action of these forces is compensated. If several bodies interact with each other only by gravitational and elastic forces and no external forces act on them, then for any interactions of bodies, the work of the elastic or gravitational forces is equal to the change in the potential energy of the bodies, taken with the opposite sign:
\(~A = -(E_(p2) - E_(p1))\) . (17)
According to the kinetic energy theorem, the work done by the same forces is equal to the change in kinetic energy:
\(~A = E_(k2) - E_(k1)\) . (18)
From a comparison of equalities (17) and (18) it is clear that the change in the kinetic energy of bodies in a closed system is equal in absolute value to the change in the potential energy of the system of bodies and opposite in sign:
\(~E_(k2) - E_(k1) = -(E_(p2) - E_(p1))\) or \(~E_(k1) + E_(p1) = E_(k2) + E_(p2) \) . (19)
Law of conservation of energy in mechanical processes:
the sum of the kinetic and potential energy of the bodies that make up a closed system and interact with each other by gravitational and elastic forces remains constant.
The sum of the kinetic and potential energy of bodies is called total mechanical energy.
Let's give a simple experiment. Let's throw a steel ball up. By giving the initial speed υ inch, we will give it kinetic energy, which is why it will begin to rise upward. The action of gravity leads to a decrease in the speed of the ball, and hence its kinetic energy. But the ball rises higher and higher and acquires more and more potential energy ( E p = m∙g∙h). Thus, kinetic energy does not disappear without a trace, but is converted into potential energy.
At the moment of reaching the top point of the trajectory ( υ = 0) the ball is completely deprived of kinetic energy ( E k = 0), but at the same time its potential energy becomes maximum. Then the ball changes direction and moves downward with increasing speed. Now the potential energy is converted back into kinetic energy.
The law of conservation of energy reveals physical meaning concepts work:
the work of gravitational and elastic forces, on the one hand, is equal to an increase in kinetic energy, and on the other hand, to a decrease in the potential energy of bodies. Therefore, work is equal to energy converted from one type to another.
Mechanical Energy Change Law
If a system of interacting bodies is not closed, then its mechanical energy is not conserved. The change in mechanical energy of such a system is equal to the work of external forces:
\(~A_(vn) = \Delta E = E - E_0\) . (20)
Where E And E 0 – total mechanical energies of the system in the final and initial states, respectively.
An example of such a system is a system in which, along with potential forces, non-potential forces act. Non-potential forces include friction forces. In most cases, when the angle between the friction force F r body is π radians, the work done by the friction force is negative and equal to
\(~A_(tr) = -F_(tr) \cdot s_(12)\) ,
Where s 12 – body path between points 1 and 2.
Frictional forces during the movement of a system reduce its kinetic energy. As a result of this, the mechanical energy of a closed non-conservative system always decreases, turning into the energy of non-mechanical forms of motion.
For example, a car moving along a horizontal section of the road, after turning off the engine, travels some distance and stops under the influence of friction forces. The kinetic energy of the forward motion of the car became zero, but the potential energy did not increase. When the car was braking, the brake pads, car tires and asphalt heated up. Consequently, as a result of the action of friction forces, the kinetic energy of the car did not disappear, but turned into internal energy of thermal motion of molecules.
Law of conservation and transformation of energy
In any physical interaction, energy is transformed from one form to another.
Sometimes the angle between the friction force F tr and elementary displacement Δ r is equal to zero and the work done by the friction force is positive:
\(~A_(tr) = F_(tr) \cdot s_(12)\) ,
Example 1. Let the external force F acts on the block IN, which can slide on the cart D(Fig. 5). If the cart moves to the right, then the work done by the sliding friction force F tr2 acting on the cart from the side of the block is positive:
Example 2. When a wheel rolls, its rolling friction force is directed along the movement, since the point of contact of the wheel with the horizontal surface moves in the direction opposite to the direction of movement of the wheel, and the work of the friction force is positive (Fig. 6):
Literature
- Kabardin O.F. Physics: Reference. materials: Textbook. manual for students. – M.: Education, 1991. – 367 p.
- Kikoin I.K., Kikoin A.K. Physics: Textbook. for 9th grade. avg. school – M.: Prosveshchenie, 1992. – 191 p.
- Elementary physics textbook: Proc. allowance. In 3 volumes / Ed. G.S. Landsberg: vol. 1. Mechanics. Heat. Molecular physics. – M.: Fizmatlit, 2004. – 608 p.
- Yavorsky B.M., Seleznev Yu.A. A reference guide to physics for those entering universities and self-education. – M.: Nauka, 1983. – 383 p.
Kinetic energy.
An integral property of matter is movement. Various forms of motion of matter are capable of mutual transformations, which, as established, occur in strictly defined quantitative ratios. The single measure of various forms of motion and types of interaction of material objects is energy.
Energy depends on the parameters of the system state, ᴛ.ᴇ. such physical quantities that characterize some essential properties of the system. Energy that depends on two vector parameters characterizing the mechanical state of the system, namely, radius vector, which determines the position of one body relative to another, and speed, which determines the speed of movement of the body in space, is called mechanical.
In classical mechanics, it seems possible to split mechanical energy into two terms, each of which depends on only one parameter:
where is the potential energy, depending on the relative location of the interacting bodies; - kinetic energy, depending on the speed of movement of the body in space.
The mechanical energy of macroscopic bodies can only change due to work.
Let us find an expression for the kinetic energy of the translational motion of a mechanical system. It is worth saying that to begin with, let’s consider a material point with mass m. Let us assume that its speed at some point in time t equal to . Let us determine the work of the resultant force acting on a material point for some time:
Considering that based on the definition of the scalar product
where is the initial and is the final speed of the point.
Magnitude
It is customary to call it the kinetic energy of a material point.
Using this concept, relation (4.12) will be written in the form
From (4.14) it follows that energy has the same dimension as work and therefore is measured in the same units.
In other words, the work resulting from all forces acting on a material point is equal to the increment of the kinetic energy of this point. Note that the increase in kinetic energy can be positive or negative depending on the sign of the work done (a force can either accelerate or retard the movement of a body). This statement is usually called the kinetic energy theorem.
The obtained result can be easily generalized to the case of translational motion of an arbitrary system of material points. The kinetic energy of a system is usually called the sum of the kinetic energies of the material points of which this system consists. As a result of adding relations (4.13) for each material point of the system, we again obtain formula (4.13), but for a system of material points:
Where m– the mass of the entire system.
Note that there is a significant difference between the theorem on kinetic energy (the law on the change in kinetic energy) and the law on the change in the momentum of the system. As is known, the increment in the momentum of a system is determined only by external forces. Due to the equality of action and reaction, internal forces do not change the impulse of the system. This is not the case with kinetic energy. The work done by internal forces, generally speaking, does not vanish. For example, when two material points move, interacting with each other by forces of attraction, each of the forces will do positive work, and the increase in kinetic energy of the entire system will be positive. Consequently, the increase in kinetic energy is determined by the work of not only external, but also internal forces.
A line integral of the 2nd kind, the calculation of which is, as a rule, simpler than the calculation of a curvilinear integral of the 1st kind.
The power of a force is the work done by a force per unit time.Since in an infinitesimal time dt the force does work dA = fsds = fdr, then the power...Set body weight values using the slidersm, plane inclination angle a , , external forceSince in an infinitesimal time dt the force does work dA = fsds = fdr, then the power...F ext A friction coefficient
and acceleration
indicated in Table 1 for your team.
At the same time, turn on the stopwatch and press the "Start" button. Stop the stopwatch when your body stops at the end of the inclined plane.
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Do this experiment 10 times and record the results of measuring the time the body slides from the inclined plane in the table. 2. |
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D) - work of friction force on the descent section;the work of the resultant forces applied to the body is equal to the change in the kinetic energy of the body.
Since the change in kinetic energy is equal to the work done by the force (3), the kinetic energy of the body is expressed in the same units as the work, i.e., in joules.
If the initial speed of movement of a body of mass m is zero and the body increases its speed to the value υ , then the work done by the force is equal to the final value of the kinetic energy of the body:
A=Literature 2−Literature 1=m⋅υ 22−0=m⋅υ 22 .
42) Trofimova T.I. Physics course. Chapter 3, §§12,13.
Ek
Potential fields A(M Potential field conservative field, a vector field whose circulation along any closed trajectory is zero. If a force field is a force field, then this means that the work of the field forces along a closed trajectory is equal to zero. For P. p.(M) there is such a unique function A u conservative field, a vector field whose circulation along any closed trajectory is zero. If a force field is a force field, then this means that the work of the field forces along a closed trajectory is equal to zero. For P. p.)(Field potential) that
= grad (see Gradient). If a field field is given in a simply connected domain Ω, then the potential of this field can be found using the formula wherein A AM- any smooth curve connecting a fixed point from Ω with a point M, t - unit vector of tangent curve A.M. and / - arc length A.M. If A(M point-based A.) - P. p., then rot A a A(M) - = 0 (see Vector field vortex). Conversely, if rot
43) Potential energy
Potential energy- a scalar physical quantity that characterizes the ability of a certain body (or material point) to do work due to its location in the field of action of forces. Another definition: potential energy is a function of coordinates, which is a term in the Lagrangian system and describes the interaction of the elements of the system. The term "potential energy" was coined in the 19th century by Scottish engineer and physicist William Rankine.
The SI unit of energy is the Joule.
Potential energy is assumed to be zero for a certain configuration of bodies in space, the choice of which is determined by the convenience of further calculations. The process of choosing this configuration is called normalization of potential energy.
A correct definition of potential energy can only be given in a field of forces, the work of which depends only on the initial and final position of the body, but not on the trajectory of its movement. Such forces are called conservative.
Also, potential energy is a characteristic of the interaction of several bodies or a body and a field.
Any physical system tends to a state with the lowest potential energy.
The potential energy of elastic deformation characterizes the interaction between parts of the body.
The potential energy in the Earth's gravitational field near the surface is approximately expressed by the formula:
Where E p- potential energy of the body, m- body mass, g- acceleration of gravity, h- the height of the center of mass of the body above an arbitrarily chosen zero level.
44) Relationship between force and potential energy
Each point of the potential field corresponds, on the one hand, to a certain value of the force vector acting on the body, and, on the other hand, to a certain value of potential energy. Therefore, there must be a certain relationship between force and potential energy.
To establish this connection, let us calculate the elementary work performed by field forces during a small displacement of the body occurring along an arbitrarily chosen direction in space, which we denote by the letter . This work is equal to
where is the projection of the force onto the direction.
Since in this case the work is done due to the reserve of potential energy, it is equal to the loss of potential energy on the axis segment:
From the last two expressions we get
The last expression gives the average value on the interval. To
to get the value at the point you need to go to the limit:
in mathematics vector,
where a is a scalar function of x, y, z, called the gradient of this scalar and denoted by the symbol . Therefore, the force is equal to the potential energy gradient taken with the opposite sign
45) Law of conservation of mechanical energy