4.1. Loss of mechanical energy and work of non-potential forces. Efficiency Cars

If the law of conservation mechanical energy was carried out in real installations (such as the Oberbeck machine), then many calculations could be done based on the equation:

T O + P O = T(t) + P(t) , (8)

Where: T O + P O = E O- mechanical energy at the initial moment of time;

T(t) + P(t) = E(t)- mechanical energy at some subsequent point in time t.

Let's apply formula (8) to the Oberbeck machine, where you can change the height of the load on the thread (the center of mass of the rod part of the installation does not change its position). We will lift the load to a height h from the lower level (where we consider P=0). Let the system with the lifted load initially be at rest, i.e. T O = 0, P O = mgh(m- mass of load on the thread). After releasing the load, movement begins in the system and its kinetic energy is equal to the sum of the energy of the translational motion of the load and the rotational motion of the rod part of the machine:

T= + , (9)

Where - speed of forward movement of the load;

, J- angular speed of rotation and moment of inertia of the rod part

For the moment of time when the load drops to the zero level, from formulas (4), (8) and (9) we obtain:

m gh=
, (10)

Where
,  0k - linear and angular velocities at the end of the descent.

Formula (10) is an equation from which (depending on the experimental conditions) the speeds can be determined And , mass m, moment of inertia J, or height h.

However, formula (10) describes the ideal type of installation, when the parts of which move, there are no friction and resistance forces. If the work done by such forces is not zero, then the mechanical energy of the system is not conserved. Instead of equation (8), in this case one should write:

T O +P O = T(t) + P(t) + A s , (11)

Where A s- the total work of non-potential forces during the entire period of movement.

For the Oberbeck machine we get:

m gh =
, (12)

Where ,  k - linear and angular velocities at the end of the descent in the presence of energy losses.

In the installation studied here, friction forces act on the axis of the pulley and the additional block, as well as atmospheric resistance forces during the movement of the load and the rotation of the rods. The work of these non-potential forces noticeably reduces the speed of movement of machine parts.

As a result of the action of non-potential forces, part of the mechanical energy is converted into other forms of energy: internal energy and radiation energy. At the same time, work As is exactly equal to the total value of these other forms of energy, i.e. The fundamental, general physical law of conservation of energy is always fulfilled.

However, in installations where the movement of macroscopic bodies occurs, mechanical energy loss, determined by the amount of work As. This phenomenon exists in all real machines. For this reason, a special concept is introduced: coefficient useful action- efficiency. This coefficient determines the ratio useful work to stored (used) energy.

In Oberbeck's machine, useful work is equal to the total kinetic energy at the end of the descent of the load onto the thread, and efficiency. is determined by the formula:

efficiency.= (13)

Here P O = mgh- stored energy consumed (converted) into kinetic energy of the machine and into energy losses equal to As, T To- total kinetic energy at the end of the load descent (formula (9)).

This video lesson is intended for self-acquaintance with the topic “The Law of Conservation of Mechanical Energy.” First, let's define total energy and a closed system. Then we will formulate the Law of Conservation of Mechanical Energy and consider in which areas of physics it can be applied. We will also define work and learn how to define it by looking at the formulas associated with it.

The topic of the lesson is one of the fundamental laws of nature - law of conservation of mechanical energy.

We previously talked about the potential and kinetic energy, and also that a body can have both potential and kinetic energy. Before talking about the law of conservation of mechanical energy, let us remember what total energy is. Total mechanical energy is the sum of the potential and kinetic energies of a body.

Also remember what is called a closed system. Closed system- this is a system in which there is a strictly defined number of bodies interacting with each other and no other bodies from the outside act on this system.

When we have defined the concept of total energy and a closed system, we can talk about the law of conservation of mechanical energy. So, the total mechanical energy in a closed system of bodies interacting with each other through gravitational forces or elastic forces (conservative forces) remains unchanged during any movement of these bodies.

We have already studied the law of conservation of momentum (LCM):

It often happens that the assigned problems can be solved only with the help of the laws of conservation of energy and momentum.

It is convenient to consider the conservation of energy using the example of a free fall of a body from a certain height. If a body is at rest at a certain height relative to the ground, then this body has potential energy. As soon as the body begins to move, the height of the body decreases, and the potential energy decreases. At the same time, speed begins to increase, and kinetic energy appears. When the body approaches the ground, the height of the body is 0, the potential energy is also 0, and the maximum will be the kinetic energy of the body. This is where the transformation of potential energy into kinetic energy is visible (Fig. 1). The same can be said about the movement of the body in reverse, from bottom to top, when the body is thrown vertically upward.

Rice. 1. Free fall of a body from a certain height

Additional task 1. “On the fall of a body from a certain height”

Problem 1

Condition

The body is at a height from the Earth's surface and begins to fall freely. Determine the speed of the body at the moment of contact with the ground.

Solution 1:

Initial speed of the body. Need to find .

Let's consider the law of conservation of energy.

Rice. 2. Body movement (task 1)

At the top point the body has only potential energy: . When the body approaches the ground, the height of the body above the ground will be equal to 0, which means that the potential energy of the body has disappeared, it has turned into kinetic energy:

According to the law of conservation of energy, we can write:

Body weight is reduced. Transforming the above equation, we obtain: .

The final answer will be: . If we substitute the entire value, we get: .

Answer: .

An example of how to solve a problem:

Rice. 3. Example of a solution to problem No. 1

This problem can be solved in another way, as vertical movement with free fall acceleration.

Solution 2 :

Let us write the equation of motion of the body in projection onto the axis:

When the body approaches the surface of the Earth, its coordinate will be equal to 0:

The gravitational acceleration is preceded by a “-” sign because it is directed against the chosen axis.

Substituting known values, we find that the body fell over time. Now let's write the equation for speed:

Assuming the free fall acceleration to be equal, we obtain:

The minus sign means that the body moves against the direction of the selected axis.

Answer: .

An example of solving problem No. 1 using the second method.

Rice. 4. Example of a solution to problem No. 1 (method 2)

Also, to solve this problem, you could use a formula that does not depend on time:

Of course, it should be noted that we considered this example taking into account the absence of friction forces, which in reality act in any system. Let's turn to the formulas and see how the law of conservation of mechanical energy is written:

Additional task 2

A body falls freely from a height. Determine at what height the kinetic energy is equal to a third of the potential energy ().

Rice. 5. Illustration for problem No. 2

Solution:

When a body is at a height, it has potential energy, and only potential energy. This energy is determined by the formula: . This will be the total energy of the body.

When a body begins to move downward, the potential energy decreases, but at the same time the kinetic energy increases. At the height that needs to be determined, the body will already have a certain speed V. For the point corresponding to the height h, the kinetic energy has the form:

The potential energy at this height will be denoted as follows: .

According to the law of conservation of energy, our total energy is conserved. This energy remains a constant value. For a point we can write the following relation: (according to Z.S.E.).

Remembering that the kinetic energy according to the conditions of the problem is , we can write the following: .

Please note: the mass and acceleration of gravity are reduced, after simple transformations we find that the height at which this relationship is satisfied is .

Answer:

Example of task 2.

Rice. 6. Formalization of the solution to problem No. 2

Imagine that a body in a certain frame of reference has kinetic and potential energy. If the system is closed, then with any change a redistribution has occurred, the transformation of one type of energy into another, but the total energy remains the same in value (Fig. 7).

Rice. 7. Law of conservation of energy

Imagine a situation where a car is moving along a horizontal road. The driver turns off the engine and continues driving with the engine turned off. What happens in this case (Fig. 8)?

Rice. 8. Car movement

In this case, the car has kinetic energy. But you know very well that over time the car will stop. Where did the energy go in this case? After all, the potential energy of the body in this case also did not change; it was some kind of constant value relative to the Earth. How did the energy change occur? In this case, the energy was used to overcome friction forces. If friction occurs in a system, it also affects the energy of that system. Let's see how the change in energy is recorded in this case.

The energy changes, and this change in energy is determined by the work against the friction force. We can determine the work of the friction force using the formula, which is known from class 7 (force and displacement are directed in opposite directions):

So, when we talk about energy and work, we must understand that each time we must take into account the fact that part of the energy is spent on overcoming friction forces. Work is being done to overcome friction forces. Work is a quantity that characterizes the change in the energy of a body.

To conclude the lesson, I would like to say that work and energy are essentially related quantities through acting forces.

Additional task 3

Two bodies - a block of mass and a plasticine ball of mass - move towards each other with the same speeds (). After the collision, the plasticine ball sticks to the block, the two bodies continue to move together. Determine what part of the mechanical energy turned into the internal energy of these bodies, taking into account the fact that the mass of the block is 3 times greater than the mass of the plasticine ball ().

Solution:

The change in internal energy can be denoted by . As you know, there are several types of energy. In addition to mechanical energy, there is also thermal, internal energy.

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One of the most important laws, according to which the physical quantity - energy is conserved in an isolated system. All known processes in nature, without exception, obey this law. In an isolated system, energy can only be converted from one form to another, but its quantity remains constant.

In order to understand what the law is and where it comes from, let’s take a body of mass m, which we drop to the Earth. At point 1, our body is at height h and is at rest (velocity is 0). At point 2 the body has a certain speed v and is at a distance h-h1. At point 3 the body has maximum speed and it almost lies on our Earth, that is, h = 0

At point 1 the body has only potential energy, since the speed of the body is 0, so the total mechanical energy is equal.

After we released the body, it began to fall. When falling, the potential energy of a body decreases, as the height of the body above the Earth decreases, and its kinetic energy increases, as the speed of the body increases. In section 1-2 equal to h1, the potential energy will be equal to

And the kinetic energy will be equal at that moment ( - the speed of the body at point 2):

The closer a body becomes to the Earth, the less its potential energy, but at the same moment the speed of the body increases, and because of this, kinetic energy. That is, at point 2 the law of conservation of energy works: potential energy decreases, kinetic energy increases.

At point 3 (on the surface of the Earth), the potential energy is zero (since h = 0), and the kinetic energy is maximum (where v3 is the speed of the body at the moment of falling to the Earth). Since , the kinetic energy at point 3 will be equal to Wk=mgh. Consequently, at point 3 the total energy of the body is W3=mgh and is equal to the potential energy at height h. The final formula for the law of conservation of mechanical energy will be:

The formula expresses the law of conservation of energy in a closed system in which only conservative forces act: the total mechanical energy of a closed system of bodies interacting with each other only by conservative forces does not change with any movements of these bodies. Only mutual transformations of the potential energy of bodies into their kinetic energy and vice versa occur.

In Formula we used.

1. Consider the free fall of a body from a certain height h relative to the Earth's surface (Fig. 77). At the point A the body is motionless, therefore it has only potential energy. At the point B on high h 1 the body has both potential energy and kinetic energy, since the body at this point has a certain speed v 1 . At the moment of touching the surface of the Earth, the potential energy of the body is zero; it has only kinetic energy.

Thus, during the fall of a body, its potential energy decreases, and its kinetic energy increases.

Total mechanical energy E called the sum of potential and kinetic energies.

E = E n + E To.

2. Let us show that the total mechanical energy of a system of bodies is conserved. Let us consider once again the fall of a body onto the surface of the Earth from a point A exactly C(see Fig. 78). We will assume that the body and the Earth represent a closed system of bodies in which only conservative forces act, in this case gravity.

At the point A the total mechanical energy of a body is equal to its potential energy

E = E n = mgh.

At the point B the total mechanical energy of the body is equal to

E = E p1 + E k1.
E n1 = mgh 1 , E k1 = .

Then

E = mgh 1 + .

Body speed v 1 can be found using the kinematics formula. Since the movement of a body from a point A exactly B equals

s = h – h 1 = , then = 2 g(h – h 1).

Substituting this expression into the formula for total mechanical energy, we get

E = mgh 1 + mg(h – h 1) = mgh.

Thus, at the point B

E = mgh.

At the moment of touching the surface of the Earth (point C) the body has only kinetic energy, therefore, its total mechanical energy

E = E k2 = .

The speed of the body at this point can be found using the formula = 2 gh, taking into account that the initial speed of the body is zero. After substituting the expression for speed into the formula for total mechanical energy, we obtain E = mgh.

Thus, we obtained that at the three considered points of the trajectory, the total mechanical energy of the body is equal to the same value: E = mgh. We will arrive at the same result by considering other points of the body’s trajectory.

The total mechanical energy of a closed system of bodies, in which only conservative forces act, remains unchanged during any interactions of the bodies of the system.

This statement is the law of conservation of mechanical energy.

3. In real systems, friction forces act. Thus, when a body falls freely in the example considered (see Fig. 78), the force of air resistance acts, therefore the potential energy at the point A more total mechanical energy at a point B and at the point C by the amount of work done by the force of air resistance: D E = A. In this case, the energy does not disappear; part of the mechanical energy is converted into the internal energy of the body and air.

4. As you already know from the 7th grade physics course, to facilitate human labor, various machines and mechanisms are used, which, having energy, perform mechanical work. Such mechanisms include, for example, levers, blocks, cranes, etc. When work is performed, energy is converted.

Thus, any machine is characterized by a quantity that shows what part of the energy transferred to it is used usefully or what part of the perfect (total) work is useful. This quantity is called efficiency(efficiency).

The efficiency h is a value equal to the ratio of useful work A n to full work A.

Efficiency is usually expressed as a percentage.

h = 100%.

5. Example of problem solution

A parachutist weighing 70 kg separated from the motionless hanging helicopter and, having flown 150 m before the parachute opened, acquired a speed of 40 m/s. What is the work done by air resistance?

Given:	Solution
m= 70 kg v 0 = 0 v= 40 m/s sh= 150 m	For the zero level of potential energy, we choose the level at which the parachutist acquired speed v. Then, when separated from the helicopter in the initial position at altitude h the total mechanical energy of a skydiver is equal to his potential energy E=E n = mgh, since its kinetic
A?

ical energy at a given altitude is zero. Having flown the distance s= h, the parachutist acquired kinetic energy, and his potential energy at this level became zero. Thus, in the second position, the total mechanical energy of the parachutist is equal to his kinetic energy:

E = E k = .

Potential energy of a skydiver E n when separated from the helicopter is not equal to the kinetic E k, since the force of air resistance does work. Hence,

A = E To - E P;

A =– mgh.

A=– 70 kg 10 m/s 2,150 m = –16,100 J.

The work has a minus sign because it is equal to the loss of total mechanical energy.

Answer: A= –16,100 J.

Self-test questions

1. What is called total mechanical energy?

2. Formulate the law of conservation of mechanical energy.

3. Is the law of conservation of mechanical energy satisfied if a friction force acts on the bodies of the system? Explain your answer.

4. What does efficiency show?

Task 21

1. A ball of mass 0.5 kg is thrown vertically upward at a speed of 10 m/s. What is the potential energy of the ball at its highest point?

2. An athlete weighing 60 kg jumps from a 10-meter platform into the water. What is equal to: the potential energy of the athlete relative to the surface of the water before the jump; its kinetic energy upon entering water; its potential and kinetic energy at a height of 5 m relative to the surface of the water? Neglect air resistance.

3. Determine the efficiency of an inclined plane 1 m high and 2 m long when a load weighing 4 kg moves along it under the influence of a force of 40 N.

Chapter 1 highlights

1. Types of mechanical movement.

2. Basic kinematic quantities (Table 2).

table 2

Name	Designation	What characterizes	Unit	Measuring method	Vector or scalar	Relative or absolute
Coordinate a	x, y, z	body position	m	Ruler	Scalar	Relative
Path	l	change in body position	m	Ruler	Scalar	Relative
Moving	s	change in body position	m	Ruler	Vector	Relative
Time	t	process duration	With	Stopwatch	Scalar	Absolute
Speed	v	speed of position change	m/s	Speedometer	Vector	Relative
Acceleration	a	speed of change of speed	m/s2	Accelerometer	Vector	Absolute

3. Basic equations of motion (Table 3).

Table 3

	Straightforward		Uniform around the circumference
	Uniform	Uniformly accelerated
Acceleration	a = 0	a= const; a =	a = ; a= w2 R
Speed	v = ; vx =	v = v 0 + at; vx = v 0x + axt	v= ; w =
Moving	s = vt; sx=vxt	s = v 0t + ; sx=vxt+
Coordinate	x = x 0 + vxt	x = x 0 + v 0xt +

4. Basic traffic schedules.

Table 4

Type of movement	Acceleration modulus and projection	Modulus and velocity projection	Module and displacement projection	Coordinate*	Path*
Uniform
Uniformly accelerated e

5. Basic dynamic quantities.

Table 5

Name	Designation	Unit	What characterizes	Measuring method	Vector or scalar	Relative or absolute
Weight	m	kg	Inertia	Interaction, weighing on lever scales	Scalar	Absolute
Force	F	N	Interaction	Weighing on spring scales	Vector	Absolute
Body impulse	p = m v	kgm/s	Body condition	Indirect	Vector	I'm relative
Impulse force	Ft	NS	Change in body state (change in body momentum)	Indirect	Vector	Absolute

6. Basic laws of mechanics

Table 6

Name	Formula	Note	Limits and conditions of applicability
Newton's first law		Establishes the existence of inertial frames of reference	Valid: in inertial reference systems; for material points; for bodies moving at speeds much lower than the speed of light
Newton's second law	a =	Allows you to determine the force acting on each of the interacting bodies
Newton's third law	F 1 = F 2	Refers to both interacting bodies
Newton's second law (other formulation)	mvm v 0 = Ft	Sets the change in the momentum of a body when an external force acts on it
Law of conservation of momentum	m 1 v 1 + m 2 v 2 = = m 1 v 01 + m 2 v 02	Valid for closed systems
Law of conservation of mechanical energy	E = E k + E P	Valid for closed systems in which conservative forces act
Law of change of mechanical energy	A=D E = E k + E P	Valid for open systems in which non-conservative forces act

7. Forces in mechanics.

8. Basic energy quantities.

Table 7

Name	Designation	Units of measurement	What characterizes	Relationship with other quantities	Vector or scalar	Relative or absolute
Job	A	J	Energy measurement	A =Fs	Scalar	Absolute
Power	N	W	Speed ​​of work completion	N =	Scalar	Absolute
Mechanical energy	E	J	Ability to do work	E = E n + E To	Scalar	Relative
Potential energy	E P	J	Position	E n = mgh E n =	Scalar	Relative
Kinetic energy	E To	J	Position	E k =	Scalar	Relative
Efficiency coefficient			What part of the completed work is useful?

An absolutely inelastic impact can also be demonstrated using plasticine (clay) balls moving towards each other. If the masses of the balls m 1 and m 2, their speed before impact, then, using the law of conservation of momentum, we can write:

If the balls were moving towards each other, then together they will continue to move in the direction in which the ball with greater momentum was moving. In a particular case, if the masses and velocities of the balls are equal, then

Let us find out how the kinetic energy of the balls changes during a central absolutely inelastic impact. Since during the collision of balls between them forces act that depend not on the deformations themselves, but on their velocities, we are dealing with forces similar to friction forces, therefore the law of conservation of mechanical energy should not be observed. Due to deformation, there is a “loss” of kinetic energy, converted into thermal or other forms of energy ( energy dissipation). This “loss” can be determined by the difference in kinetic energies before and after the impact:

.

From here we get:

(5.6.3)

If the struck body was initially motionless (υ 2 = 0), then

When m 2 >> m 1 (the mass of a stationary body is very large), then almost all the kinetic energy upon impact is converted into other forms of energy. Therefore, for example, to obtain significant deformation, the anvil must be more massive than the hammer.

When then, almost all the energy is spent on the greatest possible movement, and not on residual deformation (for example, a hammer - a nail).

An absolutely inelastic impact is an example of how “loss” of mechanical energy occurs under the influence of dissipative forces.