WORK AND MECHANICAL ENERGY.

§ 1 Energy. Mechanical energy.

Types of mechanical energy. Work

Energy is called a scalar physical quantity that is a common measure of the different forms of motion.
Quantifies the energy of the system with respect to various transformations in her movements that occur as a result of the interaction of particles with each other and with external bodies. To analyze the different forms of motion introduce various forms of energy: mechanical, internal, electromagnetic, nuclear, etc.

To mechanical energy is the energy associated with the forces of gravity, elastic deformation of the body and the energy associated with the motion of the body.
Even the definition of energy in mechanics: energy is called the body's ability to do work. Energy content is determined the work that can make the body, changing its state: suspended load in the fall, a coiled spring when restoring form: a moving body on stopping. Mechanical energy of the body is called a maximum amount equal to the work that can make the body in this environment.

I. Mechanical work (Work of a constant force)

If the body under the force  makes moving , a work force that is equal to the scalar product of the force on the motion vector. The work force is a scalar quantity

The work of the horizontal component of the force F - force equals F_traction  equal:

The work of the vertical component of the force F - lifting force is F_n:

Force whose direction is perpendicular to the direction of motion of the body, no work.
The work of the friction force  is

Force directed against the pull, and negative work force is called resistance. Force perpendicular to the displacement does not change the numerical value of the speed (this force causes the body to move in a circle - the centripetal force) and the work it is 0.
Force, increasing the numerical value of the velocity (the angle α - acute), performs positive work. Strength, which reduces the numerical value of the velocity (the angle α - obtuse), performs negative work.

II The work force of gravity. Conservative forces.

Determine the work force of gravity when moving body of mass m on an inclined plane, whose length l, and the height h. The body of two forces: gravity, directed vertically downward and reaction force , directed perpendicular to the plane of the AC. Their net  one does work to the body acceleration (force of friction is neglected).

From

b) Determine the work done by the force of gravity in free fall at the height of the body

A comparison of the work done by the force of gravity when driving on an inclined plane and the free fall shows that the work force of gravity does not depend on the length and shape of the path traversed by the body, and is determined by the product of the force of gravity at the height difference in the initial and final position.
When driving down the gravity performs positive work, you move up - negative. The work force of gravity along a closed path 1-2-1 is 0.

Force, work which does not depend on the shape and length of the path, and is determined only by the initial and final positions of the body, called conservative.

The work of conservative forces in a closed path is zero. Examples of conservative forces: gravity, the force of the spring, and the strength of the electrostatic interaction.

III. The work of friction. Dissipative forces.

The friction force F_fr determined by the relative speed of the bodies in contact (sliding friction). The friction force is always directed against the movement (α = 180 °); always a force of resistance and, therefore, of its work is always negative and after the return of the body to its original position the total work force of friction is different from 0 and negative.

Dissipative forces are called force, the total work is in any movement of a closed system is always negative. Example: the force of friction and the resistance movement of bodies in liquids and gases. As a result of the dissipative forces the mechanical energy is transformed into other forms of energy.

IV Work variable force.

Determine the work force, the magnitude of which varies from point to point, according to the law in the figure. We split S move into elementary areas dS, where the force is constant, then the elementary work written as

A complete work on all moving from point 1 to point 2 is

or, passing to the limit,

The work variable force is:

The work of the elastic force, given that

The work of the elastic force of a closed path 1-2-1

 elastic force – conservative.

 V Kinetic energy.

If elementary moving  written as

then

and

According to II Newton's law

then

Value W_k equal to   is the kinetic energy

The work of the resultant of all the forces acting on the particle is equal to the change in kinetic energy.

Then

or other record

If A > 0, the W_K increases (falls)

If A > 0, then decreases W_K (throwing).

Moving bodies have the ability to do the job and in the event that no force on the part of other bodies do not act on them. If the body is moving with constant velocity, - the sum of all the forces acting on a body is equal to 0 and the work is not done. If the body is to act with some force in the direction of the body to another, then it can do the work. According to III rd Newton's law a moving body will be attached to the same magnitude force, but in opposite directions. Due to the action of this force will decrease the speed of the body until it stops. W_K energy due to the movement of the body is called kinetic energy. Completely stop the body can not make work. W_K depends on the speed and weight. Changing the direction of the velocity does not affect the kinetic energy.

VI. Potential energy.

If the body is lifted to a height h, then fell under the influence of gravity, the body can actually work

If you press the spring on the value of X₂ = X (X₁ = 0), then return to its original state deformed spring is able to perform the work

Consequently, these bodies have stored energy, which arises due to the interaction of bodies with one another. This energy is called potential energy. Potential energy is called energy, which depends on the relative positions of the particles of the system.

If a body falls from a certain height h₁do height h₂, the potential energy changes from

to

Perfect for this work is

i.e. work performed by the bodies to which action conservative forces equals the change in potential energy of the opposite sign.

Thus, when a falling body performs positive work, his W_p decreases. If the body is lifted up, the force of gravity and makes a negative work W_p increases.

VII. Total energy.

Mechanical energy or the total mechanical energy is called the energy of mechanical motion and interaction. Mechanical energy is the sum of kinetic and potential energy.