Complete mechanical energy characterizes the movement and interaction of bodies, therefore, depends on the velocities and relative position of the bodies.

The total mechanical energy of a closed mechanical system is equal to the sum of the kinetic and potential energies of the bodies of this system:

Law of energy conservation

The law of conservation of energy is a fundamental law of nature.

In Newtonian mechanics, the law of conservation of energy is formulated as follows:

The total mechanical energy of an isolated (closed) system of bodies remains constant.

In other words:

Energy does not arise from nothing and does not disappear anywhere, it can only pass from one form to another.

Classical examples of this statement are: a spring pendulum and a pendulum on a thread (with negligible damping). In the case of a spring pendulum, in the process of oscillation, the potential energy of a deformed spring (having a maximum in the extreme positions of the load) is converted into the kinetic energy of the load (reaching a maximum at the moment the load passes the equilibrium position) and vice versa. In the case of a pendulum on a thread, the potential energy of the load is converted into kinetic energy and vice versa.

2 Equipment

2.1 Dynamometer.

2.2 Laboratory stand.

2.3 Load weighing 100 g - 2 pcs.

2.4 Measuring ruler.

2.5 Piece soft tissue or felt.

3 Theoretical background

The scheme of the experimental setup is shown in Figure 1.

The dynamometer is fixed vertically in the foot of the tripod. A piece of soft cloth or felt is placed on a tripod. When hanging loads from the dynamometer, the tension of the dynamometer spring is determined by the position of the pointer. In this case, the maximum elongation (or static displacement) of the spring X 0 occurs when the elastic force of a spring with stiffness k balances the force of gravity of the load with the mass t:

kx 0 =mg, (1)

where g = 9.81 - free fall acceleration.

Consequently,

The static displacement characterizes the new equilibrium position O" of the lower end of the spring (Fig. 2).

If the load is pulled down a distance BUT from point O" and release at point 1, then periodic oscillations of the load occur. At points 1 and 2, called turning points, the load stops, reversing the direction of movement. Therefore, at these points, the speed of the load v = 0.

Max Speed v m ax the load will have at the midpoint O". Two forces act on the oscillating load: the constant force of gravity mg and variable elastic force kx. Potential energy of a body in a gravitational field at an arbitrary point with coordinate X is equal to mgx. The potential energy of the deformed body, respectively, is equal to .

In this case, the point X = 0, corresponding to the position of the pointer for an unstretched spring.

The total mechanical energy of the load at an arbitrary point is the sum of its potential and kinetic energy. Neglecting the forces of friction, we use the law of conservation of total mechanical energy.

Let us equate the total mechanical energy of the load at point 2 with the coordinate -(X 0 -BUT) and at point O" with coordinate -X 0 :

Expanding the brackets and performing simple transformations, we bring formula (3) to the form

Then the module of the maximum speed of loads

The stiffness of a spring can be found by measuring the static displacement X 0 . As follows from formula (1),

Energy is the reserve of the system's operability. Mechanical energy is determined by the speeds of movements of bodies in the system and their mutual arrangement; hence, it is the energy of movement and interaction.

The kinetic energy of a body is the energy of its mechanical movement, which determines the ability to do work. In translational motion, it is measured by half the product of the mass of the body and the square of its speed:

At rotary motion the kinetic energy of the body has the expression:

The potential energy of a body is the energy of its position, due to the mutual relative position of bodies or parts of the same body and the nature of their interaction. Potential energy in the field of gravity:

where G is the force of gravity, h is the difference between the levels of the initial and final positions above the Earth (relative to which the energy is determined). Potential energy of elastically deformed body:

where C is the modulus of elasticity, delta l is the deformation.

Potential energy in the field of gravity depends on the location of the body (or system of bodies) relative to the Earth. The potential energy of an elastically deformed system depends on the relative arrangement of its parts. Potential energy arises due to kinetic energy (lifting the body, stretching the muscle) and when changing position (falling the body, shortening the muscle), it passes into kinetic energy.

The kinetic energy of the system during plane-parallel motion is equal to the sum of the kinetic energy of its CM (assuming that the mass of the entire system is concentrated in it) and the kinetic energy of the system in its rotational motion relative to the CM:

The total mechanical energy of the system is equal to the sum of the kinetic and potential energy. In the absence of external forces, the total mechanical energy of the system does not change.

Change in kinetic energy material system on a certain path is equal to the sum of the work of external and internal forces on the same path:

The kinetic energy of the system is equal to the work of the braking forces that will be produced when the system's velocity decreases to zero.

In human movements, one type of movement passes into another. At the same time, energy as a measure of the motion of matter also passes from one form to another. So, the chemical energy in the muscles is converted into mechanical energy (internal potential of elastically deformed muscles). The traction force of the muscles generated by the latter does the work and transforms potential energy into the kinetic energy of the moving parts of the body and external bodies. The mechanical energy of external bodies (kinetic) is transferred during their action on the human body to the links of the body, is converted into potential energy of stretched antagonist muscles and into dissipated thermal energy (see Chapter IV).

The law of conservation of energy states that the energy of the body never disappears and does not reappear, it can only turn from one form to another. This law is universal. It has its own formulation in various branches of physics. Classical mechanics considers the law of conservation of mechanical energy.

Total mechanical energy closed system physical bodies, between which conservative forces act, is a constant value. This is how the law of conservation of energy in Newtonian mechanics is formulated.

Closed, or isolated, is considered to be physical system, which is not affected by external forces. It does not exchange energy with the surrounding space, and its own energy, which it possesses, remains unchanged, that is, it is preserved. In such a system, only internal forces, and bodies interact with each other. It can only convert potential energy into kinetic energy and vice versa.

The simplest example of a closed system is a sniper rifle and a bullet.

Types of mechanical forces

The forces that act inside a mechanical system are usually divided into conservative and non-conservative.

conservative forces are considered whose work does not depend on the trajectory of the body to which they are applied, but is determined only by the initial and final position of this body. The conservative forces are also called potential. The work of such forces in a closed loop is zero. Examples of conservative forces − gravity force, elastic force.

All other forces are called non-conservative. These include friction force and drag force. They are also called dissipative forces. These forces perform negative work during any motions in a closed mechanical system, and under their action the total mechanical energy of the system decreases (dissipates). It passes into other, non-mechanical types of energy, for example, into heat. Therefore, the law of conservation of energy in a closed mechanical system can be fulfilled only if there are no non-conservative forces in it.

The total energy of a mechanical system consists of kinetic and potential energy and is their sum. These types of energies can transform into each other.

Potential energy

Potential energy is called the energy of interaction of physical bodies or their parts with each other. It is determined by their mutual arrangement, that is, the distance between them, and is equal to the work that needs to be done to move the body from the reference point to another point in the field of conservative forces.

Potential energy has any motionless physical body, raised to some height, since it is affected by gravity, which is a conservative force. Such energy is possessed by water at the edge of a waterfall, a sled at the top of a mountain.

Where did this energy come from? While the physical body was being raised to a height, work was done and energy was expended. It is this energy that was stored in the raised body. And now this energy is ready to do work.

The value of the potential energy of the body is determined by the height at which the body is located relative to some initial level. We can take any point we choose as a starting point.

If we consider the position of the body relative to the Earth, then the potential energy of the body on the surface of the Earth is zero. And on top h it is calculated by the formula:

E p = mɡ h ,

where m - body mass

ɡ - acceleration of gravity

h – height of the center of mass of the body relative to the Earth

ɡ \u003d 9.8 m / s 2

When a body falls from a height h1 up to height h2 gravity does work. This work is equal to the change in potential energy and has a negative value, since the magnitude of potential energy decreases as the body falls.

A = - ( E p2 - E p1) = - ∆ E p ,

where E p1 is the potential energy of the body at height h1 ,

E p2 - potential energy of a body at a height h2 .

If the body is raised to a certain height, then work is done against the forces of gravity. In this case, it has a positive value. And the value of the potential energy of the body increases.

An elastically deformed body (compressed or stretched spring). Its value depends on the stiffness of the spring and on how long it was compressed or stretched, and is determined by the formula:

E p \u003d k (∆x) 2 / 2 ,

where k - stiffness coefficient,

∆x - lengthening or contraction of the body.

The potential energy of the spring can do work.

Kinetic energy

Translated from the Greek "kinema" means "movement". The energy that a physical body receives as a result of its movement is called kinetic. Its value depends on the speed of movement.

A soccer ball rolling across the field, a sled rolling down a mountain and continuing to move, an arrow fired from a bow - they all have kinetic energy.

If a body is at rest, its kinetic energy is zero. As soon as a force or several forces act on the body, it will begin to move. And since the body is moving, the force acting on it does work. The work of the force, under the influence of which the body from rest will go into motion and change its speed from zero to ν , is called kinetic energy body mass m .

If, at the initial moment of time, the body was already in motion, and its speed had the value v 1 , and at the end it was equal to v 2 , then the work done by the force or forces acting on the body will be equal to the increment in the kinetic energy of the body.

∆ E k = E k 2 - E k 1

If the direction of the force coincides with the direction of motion, then positive work is done, and the kinetic energy of the body increases. And if the force is directed in the direction opposite to the direction of motion, then negative work is done, and the body gives off kinetic energy.

Law of conservation of mechanical energy

Ek 1 + E p1= E k 2 + E p2

Any physical body located at some height has potential energy. But when falling, it begins to lose this energy. Where does she go? It turns out that it does not disappear anywhere, but turns into the kinetic energy of the same body.

Suppose , at some height, a load is motionlessly fixed. Its potential energy at this point is equal to the maximum value. If we let it go, it will start falling at a certain speed. Therefore, it will begin to acquire kinetic energy. But at the same time, its potential energy will begin to decrease. At the point of impact, the kinetic energy of the body will reach a maximum, and the potential energy will decrease to zero.

The potential energy of a ball thrown from a height decreases, while the kinetic energy increases. Sledges at rest on top of a mountain have potential energy. Their kinetic energy at this moment is zero. But when they start to roll down, the kinetic energy will increase, and the potential energy will decrease by the same amount. And the sum of their values ​​will remain unchanged. The potential energy of an apple hanging on a tree is converted into its kinetic energy when it falls.

These examples clearly confirm the law of conservation of energy, which says that the total energy of a mechanical system is a constant value . Value full energy system does not change, and potential energy is converted into kinetic energy and vice versa.

By what amount the potential energy decreases, the kinetic energy will increase by the same amount. Their amount will not change.

For a closed system of physical bodies, the equality
E k1 + E p1 = E k2 + E p2,
where E k1 , E p1 - kinetic and potential energies of the system before any interaction, E k2 , E p2 - corresponding energies after it.

The process of converting kinetic energy into potential energy and vice versa can be seen by watching a swinging pendulum.

Click on the picture

Being in the extreme right position, the pendulum seems to freeze. At this moment, its height above the reference point is maximum. Therefore, the potential energy is also maximum. And the kinetic is zero, since it does not move. But the next moment the pendulum starts moving down. Its speed increases, and, therefore, its kinetic energy increases. But as the height decreases, so does the potential energy. At the bottom point, it will become equal to zero, and the kinetic energy will reach its maximum value. The pendulum will pass this point and begin to rise up to the left. Its potential energy will begin to increase, and its kinetic energy will decrease. Etc.

To demonstrate the transformation of energy, Isaac Newton invented mechanical system, which is called Newton's cradle or Newton's balls .

Click on the picture

If you deflect and then release the first ball, then its energy and momentum will be transferred to the last one through three intermediate balls, which will remain motionless. And the last ball will deflect with the same speed and rise to the same height as the first one. Then the last ball will transfer its energy and momentum through the intermediate balls to the first one, and so on.

A ball laid aside has the maximum potential energy. Its kinetic energy at this moment is zero. Having started moving, it loses potential energy and acquires kinetic energy, which reaches its maximum at the moment of collision with the second ball, and potential energy becomes equal to zero. Further, the kinetic energy is transferred to the second, then the third, fourth and fifth balls. The latter, having received kinetic energy, begins to move and rises to the same height at which the first ball was at the beginning of the movement. Its kinetic energy at this moment is equal to zero, and the potential energy is equal to the maximum value. Then it starts to fall and in the same way transfers energy to the balls in reverse order.

This continues for quite a long time and could continue indefinitely if there were no non-conservative forces. But in reality, dissipative forces act in the system, under the influence of which the balls lose their energy. Their speed and amplitude gradually decrease. And eventually they stop. This confirms that the law of conservation of energy is satisfied only in the absence of non-conservative forces.

The value that equates to half of the product of the mass of a given body and the speed of this body squared is called in physics the kinetic energy of the body or the energy of action. The change or inconstancy of the kinetic or driving energy of the body for some time will be equal to the work that has been done for a given time by a certain force acting on a given body. If the work of any force along a closed trajectory of any type is equal to zero, then a force of this kind is called a potential force. The work of such potential forces will not depend on which trajectory the body is moving. Such work is determined by the initial position of the body and its final position. The starting point or zero for the potential energy can be chosen absolutely arbitrarily. The value that will be equal to the work done by the potential force to move the body from a given position to the zero point is called in physics the potential energy of the body or the energy of the state.

For various kinds forces in physics, there are various formulas for calculating the potential or stationary energy of a body.

The work done by potential forces will be equal to the change in this potential energy, which must be taken in the opposite sign.

If you add the kinetic and potential energy of the body, you get a value called the total mechanical energy of the body. In a position where a system of several bodies is conservative, the law of conservation or constancy of mechanical energy is valid for it. A conservative system of bodies is such a system of bodies that is subject to the action of only those potential forces that do not depend on time.

The law of conservation or constancy of mechanical energy is as follows: "During any processes that occur in a certain system of bodies, its total mechanical energy always remains unchanged." Thus, the total or all mechanical energy of any body or any system of bodies remains constant if this system of bodies is conservative.

The law of conservation or constancy of total or all mechanical energy is always invariant, that is, its form of writing does not change, even when the starting point of time is changed. This is a consequence of the law of homogeneity of time.

When dissipative forces begin to act on the system, for example, such as, then a gradual decrease or decrease in the mechanical energy of this closed system occurs. This process is called energy dissipation. A dissipative system is a system in which the energy can decrease over time. During dissipation, the mechanical energy of the system is completely converted into another. This is fully consistent with the universal law of energy. Thus, there are no completely conservative systems in nature. One or another dissipative force will necessarily take place in any system of bodies.