Theorem on the kinetic energy of the system. Open Library - an open library of educational information. Physical meaning of kinetic energy

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Two cases of transformation of the mechanical motion of a material point or system of points:

1. mechanical motion is transferred from one mechanical system to another as mechanical motion;
2. 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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An example of solving a beam bending problem
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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
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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

The kinetic energy of a material point is expressed by half the product of the mass of this point and the square of its speed.

The theorem on the kinetic energy of a material point can be expressed in three forms:

that is, the differential of the kinetic energy of a material point is equal to the elementary work of the force acting on this point;

that is, the time derivative of the kinetic energy of a material point is equal to the power of the force acting on this point:

that is, the change in the kinetic energy of a material point on a finite path is equal to the work of the force acting on a point on the same path.

Table 17. Classification of tasks

If several forces act on a point, then the right-hand sides of the equations include the work or power of the resultant of these forces, which is equal to the sum of the work or powers of all component forces.

In the case of rectilinear motion of a point, directing the axis along the straight line along which the point moves, we have:

where , since in this case the resultant of all forces applied to the point is directed along the x axis.

When applying the theorem on kinetic energy in the case of non-free motion of a material point, one must keep in mind the following: if a perfect stationary constraint is imposed on the point (the point moves along an absolutely smooth stationary surface or line), then the coupling reaction is not included in the equations, because this reaction is directed along normal to the trajectory of the point and, therefore, its work is equal to zero. If we have to take friction into account, then the work or power of the friction force will enter into the kinetic energy equation.

The tasks related to this paragraph can be divided into two main types.

I. Problems on the application of the theorem on kinetic energy for the rectilinear motion of a point.

II. Problems on the application of the theorem on kinetic energy in the curvilinear motion of a point.

In addition, tasks related to type I can be divided into three groups:

1) the force acting on a point (or the resultant of several forces) is constant, i.e., where X is the projection of the force (or resultant) onto the axis directed along the rectilinear trajectory of the point;

2) the force acting on a point (or resultant) is a function of the distance (abscissa of this point), i.e.

3) the force acting on a point (or resultant) is a function of the speed of this point, i.e.

Type II tasks can be divided into three groups:

1) the force acting on a point (or resultant) is constant both in magnitude and direction (for example, weight force);

2) the force acting on a point (or resultant) is a function of the position of this point (a function of the coordinates of the point);

3) movement of a point in the presence of resistance forces.

The kinetic energy theorem is formulated as follows. The sum of the work of all forces (conservative and non-conservative) applied to a body is equal to the increment of its kinetic energy. Using this theorem we can generalize law of conservation of mechanical energy in case of open (non-isolated) system: increment total mechanical energy system is equal work outside forces over the system.

Trajectory

A trajectory is an imaginary line described by a body when moving. Depending on the shape of the movement trajectory, there are curvilinear and rectilinear. Examples of curvilinear motion: the motion of a body thrown at an angle to the horizon (trajectory - parabola), the motion of a material point in a circle.

Friction

It occurs between two bodies in the plane of contact of their surfaces and is accompanied by dissipation (dissipation) of energy. Mechanical energy of a system in which there is friction can only decrease. The science that studies friction is called tribology. It has been experimentally established that the maximum static friction force and sliding friction force do not depend on the area of ​​contact between the bodies and are proportional to the normal pressure force pressing the surfaces against each other. The proportionality coefficient is called friction coefficient(rest or sliding).

Newton's third law

Newton's third law is a physical law, according to which the forces of interaction between two material points are equal in magnitude, opposite in direction and act along a straight line connecting these points. Like Newton's other laws, the third law is valid only for inertial reference systems. A brief statement of the third law: action equals reaction.

Third escape velocity

The third cosmic speed is the minimum speed, necessary for a spacecraft launched from Earth to overcome the gravity of the Sun and leave the solar system. If the Earth at the moment of launch were stationary and did not attract the body to itself, then the third cosmic speed would be equal to 42 km/s. Taking into account the speed of the Earth's orbital motion (30 km/s), the third escape velocity is 42-30 = 12 km/s (when launched in the direction of orbital motion) or 42+30 = 72 km/s (when launched in the opposite direction). If we also take into account the force of gravity towards the Earth, then for the third escape velocity we get values ​​from 17 to 73 km/s.

Acceleration

Acceleration is a vector quantity characterizing the speed of change speed. In arbitrary motion, acceleration is defined as the ratio of the increment in speed to the corresponding period of time. If we direct this time period to zero, we get instantaneous acceleration. This means that acceleration is the derivative of speed with respect to time. If a finite period of time Δt is considered, then the acceleration is called average. In curvilinear motion, the total acceleration is the sum of tangential (tangent) And normal acceleration.

Angular velocity

Angular velocity is a vector quantity that characterizes the rotational motion of a rigid body and is directed along the axis of rotation according to the right-hand screw rule. The average angular velocity is numerically equal to the ratio of the angle of rotation to the corresponding period of time. Taking the derivative of the rotation angle with respect to time, we obtain the instantaneous angular velocity. The SI unit of angular velocity is rad/s.

Acceleration of gravity

The acceleration of a freely falling body is the acceleration with which the body moves under the influence of gravity. The acceleration of free fall is the same for all bodies, regardless of their masses. On Earth, the acceleration of a freely falling body depends on the height above sea level and on the geographic latitude and direction towards the center of the Earth. At latitude 45 0 and at sea level, the acceleration of a freely falling body is g = 9.80665 m/s 2 . In educational problems, g = 9.81 m/s 2 is usually assumed.

Physical law

A physical law is a necessary, essential and consistently repeating connection between phenomena, processes and states of bodies. Knowledge of physical laws is the main task of physical science.

50. Physical pendulum

Physical pendulum - absolutely rigid body having an axis of rotation. In a gravitational field, a physical pendulum can oscillate around the equilibrium position, while mass systems cannot be considered concentrated at one point. The period of oscillation of a physical pendulum depends on moment of inertia body and from the distance from the axis of rotation to center of mass.

Energy (from Greek energeia - activity)

Energy is a scalar physical quantity, which is a general measure of various forms of motion of matter and a measure of the transition of the motion of matter from one form to another. Main types of energy: mechanical, internal, electromagnetic, chemical, gravitational, nuclear. Some types of energy can be converted into others in strictly defined quantities (see also Law of conservation and transformation of energy).

Thermodynamics and molecular physics

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 momentum 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.

1. The kinetic energy of a body is equal to the product of the mass of the body and the square of its speed, divided in half.

2. What is the kinetic energy theorem?

2. The work of force (resultant forces) is equal to the change in the kinetic energy of the body.

3. How does the kinetic energy of a body change if the force applied to it does positive work? Negative work?

3. The kinetic energy of a body increases if the force applied to the body does positive work and decreases if the force does negative work.

4. Does the kinetic energy of a body change when the direction of its velocity vector changes?

4. Does not change, because in the formula we have V 2.

5. Two balls of equal mass roll towards each other with equal absolute velocities on a very smooth surface. The balls collide, stop for a moment, and then move in opposite directions with the same absolute speeds. What is their total kinetic energy before the collision, at the moment of the collision, and after it?

5. Total kinetic energy before collision.