Presentation on the topic "Conservation laws. Preparation for the exam". the energy of the magnetic field of the coil is converted into the energy of the electric field of the capacitor. Potential energy of a body near the Earth's surface

The part of mechanics in which movement is studied without considering the causes that cause one or another character of movement is called kinematics.
Mechanical movement called a change in the position of a body relative to other bodies
Reference system call the body of reference, the coordinate system associated with it and the clock.
Reference body called the body, relative to which the position of other bodies is considered.
material point is called a body whose dimensions in this problem can be neglected.
trajectory called a mental line, which, during its movement, describes a material point.

According to the shape of the trajectory, the movement is divided into:
A) rectilinear- the trajectory is a straight line segment;
b) curvilinear- the trajectory is a segment of the curve.

Path- this is the length of the trajectory that the material point describes for a given period of time. This is a scalar value.
moving is a vector connecting the initial position of a material point with its final position (see Fig.).

It is very important to understand how path differs from movement. The most important difference is that the movement is a vector with the beginning at the point of departure and with the end at the destination (it does not matter at all which route this movement took). And the path is, on the contrary, a scalar value that reflects the length of the trajectory traveled.

Uniform rectilinear movement called a movement in which a material point makes the same movements for any equal intervals of time
The speed of uniform rectilinear motion called the ratio of the movement to the time for which this movement occurred:

For non-uniform motion use the concept average speed. Often the average speed is entered as a scalar value. This is the speed of such uniform motion, in which the body travels the same path in the same time as with uneven motion:

instantaneous speed called the speed of the body at a given point in the trajectory or in this moment time.
Uniformly accelerated rectilinear motion- this is a rectilinear movement in which the instantaneous speed for any equal intervals of time changes by the same amount

acceleration called the ratio of change instantaneous speed body to the time during which this change occurred:

The dependence of the body coordinate on time in uniform rectilinear motion has the form: x = x 0 + V x t, where x 0 is the initial coordinate of the body, V x is the speed of movement.
free fall called uniformly accelerated motion with constant acceleration g \u003d 9.8 m / s 2 independent of the mass of the falling body. It occurs only under the influence of gravity.

The speed in free fall is calculated by the formula:

Vertical displacement is calculated by the formula:

One of the types of movement of a material point is movement in a circle. With such a movement, the speed of the body is directed along a tangent drawn to the circle at the point where the body is located (linear speed). The position of a body on a circle can be described using a radius drawn from the center of the circle to the body. The movement of a body when moving along a circle is described by turning the radius of the circle connecting the center of the circle with the body. The ratio of the angle of rotation of the radius to the time interval during which this rotation occurred characterizes the speed of movement of the body around the circle and is called angular velocity ω:

The angular velocity is related to the linear velocity by the relation

where r is the radius of the circle.
The time it takes for a body to complete one revolution is called circulation period. The reciprocal of the period - the frequency of circulation - ν

Since with uniform motion along a circle, the velocity module does not change, but the direction of the velocity changes, with such motion there is an acceleration. He is called centripetal acceleration, it is directed along the radius to the center of the circle:

Basic concepts and laws of dynamics

The part of mechanics that studies the causes that caused the acceleration of bodies is called dynamics

Newton's first law:
There are such frames of reference with respect to which the body keeps its speed constant or is at rest if no other bodies act on it or the action of other bodies is compensated.
The property of a body to maintain a state of rest or uniform rectilinear motion with balanced external forces ah, acting on it, called inertia. The phenomenon of maintaining the speed of a body with balanced external forces is called inertia. inertial reference systems called systems in which Newton's first law is satisfied.

Galileo's principle of relativity:
in all inertial reference systems under the same initial conditions, all mechanical phenomena proceed in the same way, i.e. obey the same laws
Weight is a measure of the body's inertia
Force is a quantitative measure of the interaction of bodies.

Newton's second law:
The force acting on a body is equal to the product of the mass of the body and the acceleration imparted by this force:
$F↖(→) = m⋅a↖(→)$

The addition of forces is to find the resultant of several forces, which produces the same effect as several simultaneously acting forces.

Newton's third law:
The forces with which two bodies act on each other are located on the same straight line, are equal in magnitude and opposite in direction:
$F_1↖(→) = -F_2↖(→) $

Newton's III law emphasizes that the action of bodies on each other has the character of interaction. If body A acts on body B, then body B also acts on body A (see figure).

Or in short, the force of action is equal to the force of reaction. The question often arises: why does a horse pull a sled if these bodies interact with equal forces? This is possible only through interaction with the third body - the Earth. The force with which the hooves rest on the ground must be greater than the friction force of the sled on the ground. Otherwise, the hooves will slip and the horse will not budge.
If the body is subjected to deformation, then forces arise that prevent this deformation. Such forces are called elastic forces.

Hooke's law written in the form

where k is the stiffness of the spring, x is the deformation of the body. The "−" sign indicates that the force and deformation are directed in different directions.

When bodies move relative to each other, forces arise that impede movement. These forces are called friction forces. Distinguish between static friction and sliding friction. sliding friction force calculated according to the formula

where N is the reaction force of the support, µ is the coefficient of friction.
This force does not depend on the area of ​​the rubbing bodies. The coefficient of friction depends on the material from which the bodies are made and the quality of their surface treatment.

Friction of rest occurs when the bodies do not move relative to each other. The static friction force can vary from zero to some maximum value

Gravitational forces called the forces with which any two bodies are attracted to each other.

Law gravity:
any two bodies are attracted to each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

Here R is the distance between the bodies. The law of universal gravitation in this form is valid either for material points or for spherical bodies.

body weight called the force with which the body presses on a horizontal support or stretches the suspension.

Gravity is the force with which all bodies are attracted to the Earth:

With a fixed support, the weight of the body is equal in absolute value to the force of gravity:

If a body moves vertically with acceleration, then its weight will change.
When a body moves with an upward acceleration, its weight

It can be seen that the weight of the body is greater than the weight of the resting body.

When a body moves with downward acceleration, its weight

In this case, the weight of the body is less than the weight of the resting body.

weightlessness is called such a movement of the body, in which its acceleration is equal to the acceleration of free fall, i.e. a = g. This is possible if only one force acts on the body - the force of gravity.
artificial earth satellite is a body with a speed V1 sufficient to move in a circle around the Earth
Only one force acts on the Earth's satellite - gravity, directed towards the center of the Earth
First space velocity - this is the speed that must be reported to the body so that it revolves around the planet in a circular orbit.

where R is the distance from the center of the planet to the satellite.
For the Earth, near its surface, the first escape velocity is

1.3. Basic concepts and laws of statics and hydrostatics

A body (material point) is in a state of equilibrium if the vector sum of the forces acting on it is equal to zero. There are 3 types of balance: stable, unstable and indifferent. If, when a body is taken out of equilibrium, forces arise that tend to bring this body back, this stable balance. If forces arise that tend to take the body even further away from the equilibrium position, this precarious position; if no forces arise - indifferent(See Fig. 3).


When we are talking not about a material point, but about a body that can have an axis of rotation, then in order to achieve an equilibrium position, in addition to the zero sum of the forces acting on the body, it is necessary that algebraic sum moments of all forces acting on the body, was equal to zero.

Here d is the arm of the force. Shoulder of strength d is the distance from the axis of rotation to the line of action of the force.

Lever balance condition:
the algebraic sum of the moments of all forces rotating the body is equal to zero.
By pressure they call a physical quantity equal to the ratio of the force acting on the site perpendicular to this force to the area of ​​​​the site:

For liquids and gases is valid Pascal's law:
pressure is distributed in all directions without change.
If a liquid or gas is in the field of gravity, then each higher layer presses on the lower ones, and as the liquid or gas is immersed inside, the pressure increases. For liquids

where ρ is the density of the liquid, h is the depth of penetration into the liquid.

Homogeneous liquid in communicating vessels is set at the same level. If liquid with different densities is poured into the knees of communicating vessels, then the liquid with a higher density is installed at a lower height. In this case

The heights of the liquid columns are inversely proportional to the densities:

Hydraulic Press is a vessel filled with oil or other liquid, in which two holes are cut, closed by pistons. Pistons have different sizes. If a certain force is applied to one piston, then the force applied to the second piston turns out to be different.
Thus, the hydraulic press serves to convert the magnitude of the force. Since the pressure under the pistons must be the same, then

Then A1 = A2.
A body immersed in a liquid or gas is subjected to an upward buoyant force from the side of this liquid or gas, which is called the power of Archimedes
The value of the buoyant force is set law of Archimedes: a buoyant force acts on a body immersed in a liquid or gas, directed vertically upwards and equal to the weight of the liquid or gas displaced by the body:

where ρ liquid is the density of the liquid in which the body is immersed; V submerged - the volume of the submerged part of the body.

Body floating condition- a body floats in a liquid or gas when the buoyant force acting on the body is equal to the force of gravity acting on the body.

1.4. Conservation laws

body momentum called a physical quantity equal to the product of the mass of the body and its speed:

Momentum is a vector quantity. [p] = kg m/s. Along with the momentum of the body, they often use force impulse. It is the product of force times its duration.
The change in momentum of a body is equal to the momentum of the force acting on that body. For an isolated system of bodies (a system whose bodies interact only with each other), law of conservation of momentum: the sum of the impulses of the bodies of an isolated system before the interaction is equal to the sum of the impulses of the same bodies after the interaction.
mechanical work they call a physical quantity that is equal to the product of the force acting on the body, the displacement of the body and the cosine of the angle between the direction of the force and the displacement:

Power is the work done per unit of time.

The ability of a body to do work is characterized by a quantity called energy. Mechanical energy is divided into kinetic and potential. If a body can do work due to its motion, it is said to have kinetic energy. The kinetic energy of the translational motion of a material point is calculated by the formula

If a body can do work by changing its position relative to other bodies or by changing the position of parts of the body, it has potential energy. An example of potential energy: a body raised above the ground, its energy is calculated by the formula

where h is the height of the lift

Compressed spring energy:

where k is the spring constant, x is the absolute deformation of the spring.

The sum of potential and kinetic energy is mechanical energy. For an isolated system of bodies in mechanics, law of conservation of mechanical energy: if friction forces (or other forces leading to energy dissipation) do not act between the bodies of an isolated system, then the sum of the mechanical energies of the bodies of this system does not change (the law of conservation of energy in mechanics). If there are friction forces between the bodies of an isolated system, then during the interaction part of the mechanical energy of the bodies is transferred into internal energy.

1.5. Mechanical vibrations and waves

fluctuations are called movements that have one or another degree of repetition in time. Oscillations are called periodic if the values ​​of physical quantities that change in the process of oscillations are repeated at regular intervals.
Harmonic vibrations such oscillations are called in which the oscillating physical quantity x changes according to the law of sine or cosine, i.e.

The value A, equal to the largest absolute value of the oscillating physical quantity x, is called oscillation amplitude. The expression α = ωt + ϕ determines the value of x at a given time and is called the oscillation phase. Period T The time it takes for an oscillating body to make one complete oscillation is called. The frequency of periodic oscillations called the number of complete oscillations per unit of time:

The frequency is measured in s -1 . This unit is called hertz (Hz).

Mathematical pendulum is a material point of mass m, suspended on a weightless inextensible thread and oscillating in vertical plane.
If one end of the spring is fixed motionless, and some body of mass m is attached to its other end, then when the body is taken out of equilibrium, the spring will stretch and the body will oscillate on the spring in a horizontal or vertical plane. Such a pendulum is called a spring pendulum.

The period of oscillation of a mathematical pendulum is determined by the formula

where l is the length of the pendulum.

The period of oscillation of the load on the spring is determined by the formula

where k is the stiffness of the spring, m is the mass of the load.

Propagation of vibrations in elastic media.
A medium is called elastic if there are interaction forces between its particles. Waves is the process of propagation of oscillations in elastic media.
The wave is called transverse, if the particles of the medium oscillate in directions perpendicular to the direction of wave propagation. The wave is called longitudinal, if the oscillations of the particles of the medium occur in the direction of wave propagation.
Wavelength the distance between two nearest points oscillating in the same phase is called:

where v is the speed of wave propagation.

sound waves called waves, oscillations in which occur with frequencies from 20 to 20,000 Hz.
The speed of sound is different in different environments. The speed of sound in air is 340 m/s.
ultrasonic waves called waves, the oscillation frequency of which exceeds 20,000 Hz. ultrasonic waves are not perceived by the human ear.

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Purpose: repetition of the basic concepts, laws and formulas of conservation laws in accordance with the USE codifier.

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Conservation laws: The law of conservation of mechanical energy and the law of conservation of momentum make it possible to find solutions for the impact interaction of bodies.

An absolutely inelastic impact is such a shock interaction in which the bodies are connected (stick) to each other and move on as one body. Inelastic impact (the body "sticks" to the wall): An absolutely elastic impact is a collision in which the mechanical energy of a system of bodies is conserved. Absolutely elastic impact (the body rebounds with the same velocity) If the system of bodies is not affected by external forces from other bodies, such a system is called closed;

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Conservation laws:Momentum of the body

The physical quantity equal to the product of the mass of the body and the speed of its movement is called the momentum of the body (or momentum): The physical quantity equal to the product of the force and the time of its action is called the impulse of the force (Newton's II law): The impulse of the force is equal to the change in the momentum of the body The unit of momentum in SI is kilogram-meter per second (kg m/s). Total impulse force equal to area, which is formed by a step curve with the time axis To determine the change in momentum, it is convenient to use the momentum diagram, which depicts the momentum vectors, as well as the momentum sum vector, constructed according to the parallelogram rule

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The law of conservation of momentum: In a closed system, the vector sum of the momenta of all bodies included in the system remains constant for any interactions of the bodies of this system with each other. non-central impact 1 – impulses before impact; 2 – pulses after impact; 3 – impulse diagram. Examples of applying the law of conservation of momentum: 1. Any collision of bodies (billiard balls, cars, elementary particles etc.); 2. The movement of the balloon when air comes out of it; 3. Explosions of bodies, shots, etc.

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Conservation laws:

An absolutely inelastic impact is such a shock interaction in which the bodies are connected (stick) to each other and move on as one body. Inelastic impact (the body "sticks" to the wall): Absolutely elastic impact (the body bounces off at the same speed)

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Conservation laws: Law of conservation of momentum

The law of conservation of momentum Before interaction After interaction The law of conservation of momentum is also valid for the projections of vectors on each axis

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Conservation laws: Law of conservation of momentum - jet propulsion

When firing from a gun, recoil occurs - the projectile moves forward, and the gun rolls back. A projectile and a gun are two interacting bodies. In a rocket, during the combustion of fuel, gases heated to a high temperature are ejected from the nozzle at a high speed relative to the rocket. V is the speed of the rocket after the outflow of gases. The value is called the jet thrust

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The work A performed by a constant force is a physical quantity equal to the product of the force and displacement modules multiplied by the cosine of the angle α between the force and displacement vectors; Work is a scalar quantity. It can be positive (0° ≤ α

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Conservation Laws: Power

Power N is a physical quantity equal to the ratio of work A to the time interval t during which this work is done: B international system(SI) unit of power is called watt (W) Relationships between units of power

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Conservation Laws: Kinetic Energy

Kinetic energy is the energy of motion. A physical quantity equal to half the product of the mass of the body by the square of its speed is called the kinetic energy of the body: Theorem on kinetic energy: the work of the resultant force applied to the body is equal to the change in its kinetic energy: If the body moves at a speed v, then to stop it completely, work must be done

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Conservation laws: Potential energy

Potential energy- interaction energies of bodies Potential energy is determined by the mutual position of the bodies (for example, the position of the body relative to the Earth's surface). Forces whose work does not depend on the trajectory of the body and is determined only by the initial and final positions are called conservative. The work of conservative forces on a closed trajectory is zero. The property of conservatism is possessed by the force of gravity and the force of elasticity. For these forces, we can introduce the concept of potential energy. The friction force is not conservative. The work of the friction force depends on the length of the path.

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Conservation laws: Work of force

Gravity Work: When a body is lowered, gravity does work. The work of gravity is equal to the change in the potential energy of the body, taken with the opposite sign. The work of gravity does not depend on the shape of the trajectory The work of gravity does not depend on the choice of the zero level. The work of the elastic force: In order to stretch the spring, an external force must be applied to it, the module of which is proportional to the elongation of the spring The dependence of the module of the external force on the x coordinate is shown on the graph by a straight line The potential energy of an elastically deformed body is equal to the work of the elastic force during the transition from a given state to a state with zero deformation.

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Conservation laws: The law of conservation of mechanical energy

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 unchanged. The sum E \u003d Ek + Ep is called the total mechanical energy. If friction forces act between the bodies that make up a closed system, then mechanical energy is not conserved. Part of the mechanical energy is converted into internal energy of bodies (heating). The law of conservation and transformation of energy: in any physical interactions, energy does not arise and does not disappear. It only changes from one form to another. One of the consequences of the law of conservation and transformation of energy is the assertion that it is impossible to create a “perpetuum mobile” (perpetuummobile) - a machine that could do work indefinitely without consuming energy.

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Conservation laws: Simple mechanisms. mechanism efficiency

The main purpose of simple mechanisms: Change the force in magnitude (reduce or increase) Change the direction of the force change the force in magnitude and direction

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The main mechanisms include:

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A block is a wheel with a groove around the circumference for a rope or chain, the axis of which is rigidly attached to a wall or ceiling beam. The system of blocks and cables, designed to increase the carrying capacity, is called a chain hoist. Archimedes considered the fixed block as an equal-armed lever. There is no gain in strength, but such a block allows you to change the direction of the force, which is sometimes necessary. Archimedes took the movable block as an unequal lever, giving a gain in strength by 2 times. Moments of forces act relative to the center of rotation, which at equilibrium should be equal to the “Golden Rule” of mechanics: The block does not give a gain in work.

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Conservation Laws: Lever Equilibrium Conditions

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The arm of force is the distance from the line of action of the force to the point around which the lever can turn. The illustrations show examples to help you understand: How to determine the arm of a force.

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For a non-rotating body to be in equilibrium, it is necessary that the resultant of all forces applied to the body be equal to zero The product of the modulus of force F and the arm d is called the moment of force M In the International System of Units (SI), moments of forces are measured in newton meters (N∙m ). Forces acting on the lever and their moments. M1 = F1 d1 > 0; M2 = – F2 d2

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Different types of balance of a ball on a support. (1) - indifferent equilibrium, (2) - unstable equilibrium, (3) - stable equilibrium.

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Conservation laws: mechanism efficiency

The ratio of useful work to spent work, taken as a percentage, is called the efficiency factor - efficiency. For example, when lifting a load vertically to a certain height, the useful work is -150 J, but to gain strength, they used an inclined plane and when lifting the load, they had to overcome the friction forces of the movement of the load along the inclined plane. This work will be spent 225 J.

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Consider the tasks:

USE 2001-2010 (Demo, KIM) GIA-9 2008-2010 (Demo)

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GIA 2008 24 A bullet of mass 50 g flies vertically upwards from the barrel of a gun at a speed of 40 m/s. What is the potential energy of the bullet 4 seconds after it starts moving? Ignore air resistance.

E = Ek + Ep Ek0 =Ep0 . m∙v2 /2=mgh v2 /2g=h= v0 t – gt2/2 gt2/2 - v0 t + v2 /2g = 0 t2 - 8 t + 16 = 0 t = 4 s Ep0 =m∙v2 /2 ,Ep0 = 0.05∙402 /2 = 40 J Answer: _______________W 40 J

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(GIA 2009) 3. A body thrown vertically upward from the surface of the earth reaches highest point and falls to the ground. If air resistance is not taken into account, then the total mechanical energy of the body

the same at any moment of the body’s movement the maximum at the moment of the beginning of the movement the maximum at the moment of reaching the highest point the maximum at the moment of falling to the ground

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(GIA 2009) 22. A cart with a mass of 20 kg moving at a speed of 0.8 m/s is coupled with another cart with a mass of 30 kg moving towards it at a speed of 0.2 m/s. What is the speed of the carts after coupling, when the carts move together?

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GIA 2010 3. To give the most effective acceleration space ship the jet of exhaust gases escaping from the nozzle of his jet engine must be directed

in the direction of the ship's movement opposite to the direction of the ship's movement perpendicular to the direction of the ship's movement under arbitrary angle to the direction of the ship

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(GIA 2010) 24. The conveyor evenly lifts a load of 190 kg to a height of 9 m in 50 s. Determine the current strength in the electric motor if the voltage in the electrical network is 380 V. The efficiency of the conveyor motor is 60%.

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(GIA 2010) 25. The kettlebell falls to the ground and hits an obstacle. The speed of the weight before impact is 140 m/s. What was the temperature of the weight before the impact if the temperature rose to 1000C after the impact? Assume that the entire amount of heat released upon impact is absorbed by the weight. The specific heat capacity of the weight is 140 J/(kg 0C).

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(USE 2001, demo) A3. A car with a mass of 3000 kg is moving at a speed of 2 m/s. What is the kinetic energy of the car?

3000 J 1500 J 12000 J 6000 J

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(USE 2001) A4. In order to reduce the kinetic energy of the body by 2 times, it is necessary to reduce the speed of the body by

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(Unified State Examination 2001, Demo) A4. After burning out the thread holding the spring (see figure), the left trolley began to move at a speed of 0.4 m/s. The figure shows the masses of goods together with trolleys. With what modulo speed will the right cart move?

0.4 m/s 0.8 m/s 0.2 m/s 1.2 m/s

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(Unified State Exam 2001, Demo) A5. An object of mass m = 2 kg fell to the ground from a balcony with a height h = 3 m. The change in the energy of its gravitation towards the Earth is equal in this case. . .

6 J. 60 J. 20 J. 20/3 J.

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(USE 2001) A6. A man takes water from a well 10 m deep. The mass of the bucket is 1.5 kg, the mass of water in the bucket is 10 kg. What kind of work does a man do?

1150 J 1300 J 1000 J 850 J

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(USE 2001) A7. The ball was rolled down the hill along three different chutes. In which case is the speed of the ball at the end of the path the greatest? Ignore friction.

in the first in the second in the third in all cases the speed is the same

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(USE 2001) A8. A heavy hammer falls on the pile and drives it into the ground. In this process, the transformation

hammer potential energy into pile internal energy hammer kinetic energy into hammer internal energy, pile, soil hammer internal energy into pile kinetic and potential energy hammer internal energy into pile and soil internal energy.

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(USE 2001) A29. Two plasticine balls with masses m1 = 0.1 kg and m2 = 0.2 kg fly towards each other with speeds v1 = 20 m/s and v2 = 10 m/s. When they collide, they stick together. How much did the internal energy of the balls change during the collision?

1.9 J 2 J 3 J 4 J

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(Unified State Examination 2002, Demo) A5. A cart of mass m moving at speed v collides with a stationary cart of the same mass and engages with it. The momentum of the carts after interaction is equal to

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(USE 2002, KIM) A5. In order to reduce the kinetic energy of the body by 2 times, it is necessary to reduce the speed of the body by ...

2 times 4 times times times

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(USE 2002, Demo) A28. A load attached to a spring with a stiffness of 40 N/m performs forced vibrations. The dependence of the amplitude of these oscillations on the frequency of the driving force is shown in the figure. Determine the total energy of vibrations of the load at resonance.

10–1 J 510–2 J 1.2510–2 J 210–3J

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(USE 2003, KIM) A5. A boy threw a soccer ball weighing 0.4 kg to a height of 3 m. How much has the potential energy of the ball changed?

4 J 12 J 1.2 J 7.5 J

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(USE 2003, demo) A26. The stationary boat, together with the hunter in it, has a mass of 250 kg. The hunter fires a hunting rifle in a horizontal direction. What speed will the boat get after the shot? The mass of the bullet is 8 g, and its speed at departure is 700 m/s.

22.4 m/s 0.05 m/s 0.02 m/s 700 m/s

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(USE 2004, KIM) A5. A load with a mass of 1 kg under the action of a force of 50 N directed vertically upwards rises to a height of 3 m. The change in the kinetic energy of the load is equal to

30 J 120 J 150 J 180 J

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(USE 2004, demo) A21. A rocket with a mass of 105 kg is launched vertically upward from the Earth's surface with an acceleration of 15 m/s2. If the forces of air resistance at launch are neglected, then the thrust force of the rocket engines is equal to

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(USE 2004, demo) A22. A meteorite fell to Earth from outer space. Did the mechanical energy and momentum of the Earth-meteorite system change as a result of the collision?

both the mechanical energy of the system and its momentum have changed the momentum of the system has not changed, its mechanical energy has changed the mechanical energy of the system has not changed, its momentum has not changed

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(USE 2005, DEMO) A5. The potential energy of interaction with the Earth of a weight of 5 kg increased by 75 J. This happened as a result of the fact that the weight

raised by 1.5 m lowered by 1.5 m raised by 7 m lowered by 7 m

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(USE 2005, DEMO) A7. A body of mass 2 kg moves along the x-axis. Its coordinate changes according to the equation x = A + Bt + Ct2, where A = 2 m, B = 3 m/s, C = 5 m/s2. What is the momentum of the body at time t = 2 s?

86 kgm/s 48 kgm/s 46 kgm/s 26 kgm/s

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USE - 2006, DEMO. A 27. A boy weighing 50 kg, standing on very smooth ice, throws a load weighing 8 kg at an angle of 60o to the horizon with a speed of 5 m/s. What speed will the boy acquire?

5.8 1.36 m/s 0.8 m/s 0.4 m/s

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(USE 2006, DEMO) A26. A plasticine ball weighing 0.1 kg flies horizontally at a speed of 1 m/s (see figure). It hits a stationary trolley with a mass of 0.1 kg, attached to a light spring, and sticks to the trolley. What is the maximum kinetic energy of the system during its further oscillations? Ignore friction. The impact is considered instantaneous.

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(USE 2007, DEMO) A6. Two cars of the same mass m move with velocities v and 2v relative to the Earth in one straight line in opposite directions. What is the momentum modulus of the second car in the reference frame associated with the first car?

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(USE 2007, DEMO) A9. The speed of the thrown ball just before hitting the wall was twice its speed just after hitting it. Upon impact, an amount of heat equal to 15 J was released. Find the kinetic energy of the ball before impact.

5 J 15 J 20 J 30 J

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(USE 2008, DEMO) A6. Balls of the same mass move as shown in the figure and collide absolutely inelastically. What will be the momentum of the balls after the collision?

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(USE 2008, DEMO) A9. A plasticine ball with a mass of 0.1 kg has a speed of 1 m/s. It hits a stationary trolley with a mass of 0.1 kg attached to a spring and sticks to the trolley (see figure). What is the total mechanical energy of the system during its further vibrations? Ignore friction.

0.1 J 0.5 J 0.05 J 0.025 J

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(USE 2009, DEMO) A4. A car and a truck move at speeds υ1= 108 km/h and υ2= 54 km/h. The mass of a passenger car m = 1000 kg. What is the mass of the truck if the ratio of the momentum of the truck to the momentum of the car is 1.5?

3000 kg 4500 kg 1500 kg 1000 kg

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(USE 2009, DEMO) A5. A sled of mass m is pulled uphill at a constant speed. When the sled rises to a height h from its original position, its total mechanical energy

will not change will increase by mgh will be unknown, since the slope of the slide is not set will be unknown, since the coefficient of friction is not set

Slide 56

(USE 2010, DEMO) A4. The body moves in a straight line. Under the action of a constant force of 4 N for 2 s, the momentum of the body increased and became equal to 20 kg⋅m/s. The initial momentum of the body is

4 kg⋅m/s 8 kg⋅m/s 12 kg⋅m/s 18 kg⋅m/s

Slide 57

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Physel.ru [Text, pictures]/ http://www.physel.ru/mainmenu-4/--mainmenu-9/97-s-94----.html Andrus V.F. WORK, POWER, ENERGY [Text, pictures]/ http://www.ntpo.com/physics/opening/open2000_2/31.shtml Baldina E.A. Class! physics for the curious [Text, animation]/ http://www.yaplakal.com/forum2/topic246641.html Berkov, A.V. etc. The most complete edition standard options real tasks USE 2010, Physics [Text]: textbook for graduates. cf. textbook institutions / A.V. Berkov, V.A. Mushrooms. - OOO Astrel Publishing House, 2009. - 160 p. Pulse. Law of conservation of momentum// http://www.edu.delfa.net/CONSP Kasyanov, V.A. Physics, grade 11 [Text]: a textbook for general education schools/ V.A. Kasyanov. - LLC "Drofa", 2004. - 116 p. Moment of power. Wikipedia [text, figure]/http://ru.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D1%81%D0% B8%D0%BB%D1%8B Power. Material from Wikipedia - the free encyclopedia / [Text]: / http://ru.wikipedia.org/wiki/%D0%9C%D0%BE%D1%89%D0%BD%D0%BE%D1%81%D1 %82%D1%8C Myakishev G.Ya., Kondrasheva L., Kryukov S. Work of friction forces //Kvant. - 1991. - No. 5. - S. 37-39. Myakishev, G.Ya. etc. Physics. Grade 11 [Text]: textbook for secondary schools / textbook for secondary schools G.Ya. Myakishev, B.B. Bukhovtsev. - "Enlightenment", 2009. - 166 p. Open physics [text, pictures]/ http://www.physics.ru Preparing for the exam /http://egephizika Simple mechanisms that were a mystery, many animations [Text, animations]/ http://www.yaplakal.com /forum2/topic246641.html Forces in mechanics/ http://egephizika.26204s024.edusite.ru/DswMedia/mehanika3.htm Newton's three laws / http://rosbrs.ru/konkurs/web/2004 Federal Institute pedagogical measurements. Control measuring materials(KIM) Physics //[Electronic resource]// http://fipi.ru/view/sections/92/docs/ Shapiev I.Sh. Lesson number 52. simple mechanisms. /http://physics7.edusite.ru/p4aa1.html

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1 C1.1. After the push, the ice rolled into a pit with smooth walls, in which it can move almost without friction. The figure shows a graph of the dependence of the energy of interaction of an ice floe with the Earth on its coordinates in the pit. At some point in time, the ice floe was at point A with the coordinate x = 10 cm and moved to the left, having a kinetic energy equal to 2 J. Can the ice floe slip out of the pit? Explain your answer by indicating what physical patterns you used to explain. C1.2. After the push, the ice rolled into a pit with smooth walls, in which it can move almost without friction. The figure shows a graph of the dependence of the energy of interaction of an ice floe with the Earth on its coordinates in the pit. At some point in time, the ice floe was at point A with the coordinate x = 50 cm and moved to the left, having a kinetic energy equal to 2 J. Can the ice floe slip out of the pit? Explain your answer by indicating what physical patterns you used to explain. C2.1. C2.2. C F781 A body weighing 1 kg is thrown from the Earth's surface at a speed of 20 m/s at an angle of 45 0 to the horizon. What work was done by gravity during the flight of the body (from the throw to the fall to the ground)? Ignore air resistance. 0 С2.4. C38106 A sled with riders with a total mass of 100 kg is moving down a mountain 8 m high and 100 m long. What is the average resistance force of the sled if at the end of the mountain it reaches a speed of 10 m/s and the initial speed is zero? 30 N C2.5. A bar of mass m 1 = 600 g, moving at a speed v 1 = 2 m/s, collides with a fixed bar of mass m 2 = 200 g. What will be the speed of the first bar after the collision? The impact is assumed to be central and absolutely elastic. 1 m/s. C2.6. A bar of mass m 1 = 500 g slides down an inclined plane from a height h and, moving along a horizontal surface, collides with a fixed bar of mass m 2 = 300 g. As a result of a completely inelastic collision, the total kinetic energy of the bars becomes 2.5 J. Determine the height inclined plane h. Ignore friction during movement. Assume that the inclined plane smoothly turns into a horizontal one. h= 0.8 m. C2.7. A bar of mass m 1 = 500 g slides down an inclined plane of height h = 0.8 m and collides with a fixed bar of mass m 2 = 300 g lying on a horizontal surface. Assuming that the collision is elastic, determine the kinetic energy of the first block after the collision. Ignore friction during motion.

2 Answer 0.25 J. C2.8. On a smooth horizontal plane there is a smooth hill with a height H = 24 cm and a mass M = 1 kg, and on its top lies a small washer with a mass m = 200 g (see figure). After a slight push, the puck slides off the hill and moves perpendicular to the wall, fixed in a vertical position on the plane. With what speed is the puck approaching the wall along the plane? C2.9. A puck thrown along an inclined plane slides down it, moving up and then moving down. The plot of the puck speed modulus versus time is given in the figure. Find the angle of inclination of the plane to the horizon. = arcsin 0.125. V, m/s t, s С2.10. A bar of mass m 1 = 500 g slides down an inclined plane from a height h = 0.8 m and, moving along a horizontal surface, collides with a fixed bar of mass m 2 = 300 g. Considering the collision to be absolutely inelastic, determine the total kinetic energy of the bars after the collision. Ignore friction during motion. Assume that the inclined plane smoothly turns into a horizontal one. Ek = 2.5 J. C2.11. A bar of mass m 1 = 500 g slides down an inclined plane of height h = 0.8 m and collides with a fixed bar of mass m 2 = 300 g lying on a horizontal surface. Assuming that the collision is elastic, determine the kinetic energy of the first block after the collision. Ignore friction during motion. 0.25 J C2.12. A bar of mass m 1 = 0.5 kg slides down an inclined plane from a height h = 0.8 m and, moving along a horizontal surface, collides with a fixed bar of mass m 2 = 0.3 kg. Assuming the collision is absolutely inelastic, calculate the total kinetic energy of the bars after the collision. Ignore friction during motion. Assume that the inclined plane smoothly turns into a horizontal one. C2.13. A bar of mass m 1 = 600 g, moving at a speed v 1 = 2 m/s, collides with a fixed bar of mass m 2 = 200 g. What will be the speed of the first bar after the collision? The impact is assumed to be central and absolutely elastic. 1 m/s

3 C2.14. A block of mass m slides along the horizontal surface of the table and catches up with a block of mass 6m sliding along the table in the same direction. As a result of inelastic collision, the bars stick together. Their speeds before impact were v 0 = 7 m/s and v 0 /3. The coefficient of sliding friction between the bars and the table is μ = 0.5. How far will the sticky bars move by the moment when their speed becomes 2v o /7? 0.5 m S2.15. A washer of mass m starts moving along the chute AB from point A from a state of rest. Point A is located above point B at a height of H = 6 m. In the process of moving along the chute, the mechanical energy of the puck decreases by ΔE = 2 J due to friction. At point B, the puck flies out of the chute at an angle α = 15 to the horizon and falls to the ground at point D, which is on the same horizontal line as point B (see figure). BD \u003d 4 m. Find the mass of the washer m. Neglect air resistance. t = 0.1 kg. C2.16. A washer of mass m = 100 g starts moving along the chute AB from point A from a state of rest. Point A is located above point B at a height of H = 6 m. In the process of moving along the chute, the mechanical energy of the puck decreases by ΔE = 2 J due to friction. At point B, the puck flies out of the chute at an angle of α = 15 0 to the horizon and falls on ground at point D. located on the same horizontal line with point B (see figure). Find BD. Ignore air resistance. BD = 4 m C2.17. A washer of mass m = 100 g starts moving along the chute AB from point A from a state of rest. Point A is located above point B at a height of H = 6 m. In the process of moving along the chute, the mechanical energy of the washer decreases by ΔE due to friction. At point B, the puck flies out of the chute at an angle α = 15 to the horizon and falls to the ground at point D, which is on the same horizontal line as point B (see figure). BD = 4 m. Find the value of ΔE. Ignore air resistance. ΔE = 2 J. C2.18. CE1284 A slide with two tops, heights h and 3h, rests on a smooth horizontal table surface (see figure). On the right top of the slide there is a puck, the mass of which is 12 times less than the mass of the slide. From a slight push, the puck and the slide come into motion, and the puck moves to the left, without breaking away from the smooth surface of the slide, and the progressively moving slide does not come off the table. Find the speed of the slide when the puck reaches the left top of the slide.

4 C2.19. A small puck after impact slides up the inclined plane from point A (see figure). At point B, the inclined plane passes without a break into the outer surface of a horizontal pipe with radius R. If at point A the speed of the washer exceeds v 0 = 4 m / s, then at point B the washer breaks away from the support. The length of the inclined plane AB = L = 1 m, the angle α = 30. The coefficient of friction between the inclined plane and the washer μ = 0.2. Find the outer radius of the pipe R. 0.3 m. C2.20. A small puck after a push acquires a speed v = 2 m/s and slides along the inner surface of a smooth fixed ring with a radius R = 0.14 m. At what height h does the puck come off the ring and begin to fall freely? h 0.18m. C2.21. A piece of plasticine collides with a bar resting on a horizontal surface of the table and sticks to it. The speed of plasticine before impact is v pl \u003d 5 m / s. The mass of the bar is 4 times the mass of plasticine. The coefficient of sliding friction between the bar and the table is μ = 0.25. How far will sticky blocks with plasticine move by the moment when their speed decreases by 40%? S = m. C2.22. A piece of plasticine collides with a bar sliding towards the horizontal surface of the table and sticks to it. The velocities of plasticine and the bar before impact are directed oppositely and are equal to v pl \u003d 15 m / s and v br \u003d 5 m / s. The mass of the bar is 4 times the mass of plasticine. The coefficient of sliding friction between the bar and the table is μ = 0.17. How far will the sticky blocks with plasticine move by the moment when their speed decreases by 30%? S = 0.15 m. C2.23. A piece of plasticine collides with a bar sliding towards the horizontal surface of the table and sticks to it. The velocities of plasticine and the bar before impact are mutually opposite and equal to v pl =15 m/s and v br = 5 m/s. The mass of the bar is 4 times the mass of plasticine. The coefficient of sliding friction between the bar and the table is μ = 0.17. How far will the sticky blocks with plasticine move by the moment when their speed decreases by 2 times? S = 0.22 m. C2.24. A piece of plasticine collides with a bar sliding towards the horizontal surface of the table and sticks to it. The velocities of plasticine and the bar before impact are mutually opposite and equal to v pl = 15 m/s and v br = 5 m/s. The mass of the bar is 4 times the mass of plasticine. By the moment when the speed of the stuck together bar and plasticine decreased by 2 times, they moved by 0.22 m. Determine the coefficient of friction μ of the bar on the table surface. μ = 0.17. C2.25. A trolley with a mass of 0.8 kg moves by inertia at a speed of 2.5 m/s. A piece of plasticine weighing 0.2 kg falls vertically onto a cart from a height of 50 cm and sticks to it. Calculate the energy that was converted into internal energy during this impact. Q = 1.5 J.

5 S2.26. The bullet flies horizontally at a speed of v 0 = 150 m/s, pierces a block standing on a horizontal ice surface and continues to move in the same direction at a speed. The mass of the bar is 10 times the mass of the bullet. The coefficient of sliding friction between the bar and ice μ = 0.1. By what distance S will the block move by the moment when its speed decreases by 10%? C2.27. A bullet flying horizontally with a speed v o = 120 m/s pierces a box lying on the horizontal surface of the table and continues moving in the same direction, losing 80% of its speed. The mass of the box is 16 times the mass of the bullet. The coefficient of sliding friction between the box and the table is μ = 0.5. How far will the box move by the time its speed is halved? C2.28. From the impact of a copra with a mass of 450 kg, falling freely from a height of 5 m, a pile with a mass of 150 kg is immersed in the ground by 10 cm. Determine the resistance force of the soil, assuming it to be constant, and the impact is absolutely inelastic. Ignore the change in the potential energy of the pile in the Earth's gravitational field. C2.29. The cannon, fixed at a height of 5 m, shoots in a horizontal direction with projectiles with a mass of 10 kg. Due to the recoil, its barrel, which has a mass of 1000 kg, compresses the spring of stiffness N / m by 1 m, reloading the gun. Assuming that the relative share η = 1/6 of the recoil energy goes to compress the spring, find the range of the projectile. C2.30. A spring-loaded pistol was fired vertically downwards at a target 2 m away from it. Having done work of 0.12 J, the bullet stuck in the target. What is the mass of the bullet if the spring was compressed by 2 cm before firing, and its stiffness was 100 N/m? C2.31. A massive load lying on a horizontal plane is attached to one end of a light spring with a stiffness of k = 100 N/m, while the other end of the spring is fixed motionless (see figure). The coefficient of friction of the load along the plane μ = 0.2. The load is displaced horizontally, stretching the spring, then released with the initial speed, zero. The load moves in one direction and then stops at a position where the spring is already compressed. The maximum extension of the spring at which the load moves in this way is d = 15 cm. Find the mass m of the load. C2.32. The boat stands motionless in the water with its bow to the shore. Two fishermen, standing on the shore opposite the boat, begin to pull it up with the help of two ropes, acting on the boat with constant forces (see Fig.). If only the first fisherman had pulled the boat, she would have approached the

6 reg at a speed of 0.3 m / s, and if only the second pulled at a speed of 0.4 m / s. With what speed will the boat approach the shore when both fishermen pull it? Ignore water resistance. 0.5 m/s. C2.33. What is the average pressure of powder gases in the barrel of a gun if the speed of a projectile that has flown out of it is 1.5 km/s? Barrel length 3 m, diameter 45 mm, projectile weight 2 kg. (The friction is negligible.) p = 4, Pa. C2.34. In the "Flying Cyclist" stunt, the rider moves along the springboard under the influence of gravity, starting from rest at a height H (see figure). At the edge of the springboard, the speed of the rider is directed at such an angle to the horizon that the range of his flight is maximum. After flying through the air, the racer lands on a horizontal table at the same height as the edge of the springboard. What is the flight height h on this springboard? Ignore air resistance and friction. lift height C2.35. In the "Flying Cyclist" stunt, the rider moves along the springboard under the influence of gravity, starting from rest at a height H (see figure). At the edge of the springboard, the rider's speed is directed at an angle α = 30 to the horizon. After flying through the air, the racer lands on a horizontal table at the same height as the edge of the springboard. What is the flight range L on this ski jump? Ignore air resistance and friction. flight range С2.36. In the "Flying Cyclist" trick, the racer moves on a smooth springboard under the influence of gravity, starting from rest at a height H (see figure). At the edge of the springboard, the rider's speed is directed at an angle a = 60 to the horizon. Flying through the air, he landed on a horizontal table at the same height as the edge of the springboard. What is the flight time? flight time C2.37. The initial velocity of a projectile fired vertically upwards from a cannon is 500 m/s. At the point of maximum lift, the projectile exploded into two fragments. The first fell to the ground near the point of the shot, having a speed 2 times greater than the initial velocity of the projectile, and the second in the same place - 100 s after the break. What is the ratio of the mass of the first fragment to the mass of the second fragment? Ignore air resistance.

7 S2.38. A projectile of mass 4 kg flying at a speed of 400 m/s is torn into two equal parts, one of which flies in the direction of the projectile, and the other in opposite side. At the moment of rupture, the total kinetic energy of the fragments increased by ΔE. The speed of a fragment flying in the direction of the projectile is 900 m/s. Find ΔE. ΔE = 0.5 MJ. C2.39. A 4 kg projectile flying at a speed of 400 m/s breaks into two equal parts, one of which flies in the direction of the projectile and the other in the opposite direction. At the moment of rupture, the total kinetic energy of the fragments increased by ΔE = 0.5 MJ. Determine the speed of the fragment flying in the direction of the projectile. v 1 \u003d 900 m / s. C2.40. The projectile in flight is torn into two equal parts, one of which continues to move in the direction of the projectile, and the other in the opposite direction. At the moment of rupture, the total kinetic energy of the fragments increases due to the energy of the explosion by ΔE. The speed modulus of a fragment moving in the direction of the projectile is V 1, and the speed modulus of the second fragment is V 2. Find the mass of the projectile. C2.41. Two bodies, whose masses are respectively m 1 = 1 kg and m 2 = 2 kg, slide on a smooth horizontal table (see figure). The speed of the first body v 1 = 3 m/s, the speed of the second body v 2 = 6 m/s. How much heat will be released when they collide and move on, clinging together? There is no rotation in the system. Ignore the action of external forces. Q = 15 (J). C2.43. A projectile with a mass of 2t, moving at a speed v 0, is torn into two equal parts, one of which continues to move in the direction of the projectile, and the other in the opposite direction. At the moment of rupture, the total kinetic energy of the fragments uv 2 90 m 2 v 1 m 1 С2.42. The figure shows a photograph of the installation for studying the sliding of a carriage (1) weighing 40 g along an inclined plane at an angle of 30. At the moment the movement starts, the upper sensor (2) turns on the stopwatch (3). When the carriage passes the bottom sensor (4), the stopwatch stops. Estimate the amount of heat released when the carriage slides along the inclined plane between the sensors Q 0.03 (J). 3

8 is increased due to the energy of the explosion by the value ΔЕ. The speed of a fragment moving in the direction of the projectile is v 1. Find ΔE. C2.44. The thread of the pendulum with the length l = 1 m, to which the weight m = 0.1 kg is suspended, is deflected by an angle α from the vertical position and released. The initial speed of the load is zero. The modulus of the thread tension at the moment the pendulum passes the equilibrium position T = 2 N. What is the angle α? C2.45. An elastic ball moving along a smooth horizontal plane with a speed experiences an absolutely elastic non-frontal collision with the same ball at rest, as a result of which it continues to move with a speed directed at an angle φ = 30 0 to the original direction. At what angle α to the initial direction of motion of the first ball is the velocity of the second ball directed after the collision? C2.46. A small ball is suspended on an inextensible and weightless thread l = 0.5 m long. The ball in the equilibrium position is given a horizontal speed υ 0 = 4 m / s. Calculate the maximum height h, counting from the equilibrium position of the ball, after which the ball will stop moving in a circle of radius l. 0.7 m. C2.47. Two balls, the masses of which differ by a factor of 3, hang in contact on vertical threads (see figure). A light ball is deflected through an angle of 90 and released without initial velocity. Find the ratio of the momentum of the light ball to the momentum of the heavy ball immediately after a perfectly elastic central impact. C2.48. Two balls, the masses of which are 200 g and 600 g, respectively, hang, touching, on identical vertical threads 80 cm long. The first ball was deflected at an angle of 90 and released. To what height will the balls rise after the impact if this impact is absolutely inelastic? h = 0.05 m. C2.49. Two balls, the masses of which differ by a factor of 3, hang, touching, on vertical threads (see figure). A light ball is deflected through an angle of 90 and released without initial velocity. What will be the ratio of the kinetic energies of the heavy and light balls immediately after their perfectly elastic central impact? C2.50. A ball of mass 1 kg, suspended from a thread 90 cm long, is retracted from the equilibrium position through an angle of 60 and released. At the moment the ball passes the equilibrium position c.

It is hit by a bullet of mass 10 g flying towards the ball at a speed of 300 m/s. It breaks through it and flies out horizontally at a speed of 200 m/s, after which the ball continues to move in the same direction. What is the maximum angle the ball will deflect after being hit by a bullet? (The mass of the ball is assumed to be unchanged, the diameter of the ball is negligible compared to the length of the thread.) C2.51. A ball of mass 1 kg, suspended on a thread 90 cm long, is retracted from the equilibrium position through an angle of 60 ° and released. At the moment the ball passes the equilibrium position, a bullet of mass 10 g, flying towards the ball, hits it. She breaks through it and continues to move horizontally. Determine the change in the speed of the bullet as a result of hitting the ball if it, continuing to move in the same direction, deviates through an angle of 39 o. (The mass of the ball is assumed to be unchanged, the diameter of the ball is negligible compared to the length of the thread, cos 39 = 7 9.) 100 m/s. C2.52. A ball of mass 1 kg, suspended from a thread 90 cm long, is retracted from the equilibrium position through an angle of 60 and released. At the moment the ball passes the equilibrium position, a bullet with a mass of 10 g, flying towards the ball, hits it, it pierces it and continues to move horizontally at a speed of 200 m/s. With what speed did the bullet fly if the ball, continuing to move in the horizontal direction, deviates through an angle of 39? (The mass of the ball is considered unchanged, the diameter of the ball is negligible compared to the length of the thread, cos 39 = 7/9). 300 m/s. C2.53. The figure shows a spring pendulum 2, located vertically. The mass of the pendulum platform m 2 = 0.2 kg, the length of the spring L = 10 cm. A washer 1 with a mass m 1 = 0.1 kg falls onto the spring pendulum from a height H = 25 cm. After the collision, the platform with the puck oscillates as a whole. Calculate the energy that was converted into internal energy when the puck collided with the pendulum platform. 0.1 J. S2.54. A system of weights m and M and a light inextensible thread connecting them at the initial moment rests in a vertical plane passing through the center of the fixed sphere. The weight m is located at the point on the top of the sphere (see figure). In the course of the movement that has arisen, the load m is detached from the surface of the sphere, having passed an arc 30 along it. Find the mass M if m = 100 g. The dimensions of the load m are negligible compared to the radius of the sphere. Ignore friction. Make a schematic drawing showing the forces acting on the loads.

10 S2.55. A system of weights m and M and a light inextensible thread connecting them at the initial moment rests in a vertical plane passing through the center of the fixed sphere. The weight m is located at the point on the top of the sphere (see figure). In the course of the movement that has arisen, the load m is detached from the surface of the sphere, having passed an arc 30 along it. Find the mass M if m = 100 g. The dimensions of the load m are negligible compared to the radius of the sphere. Ignore friction. Make a schematic drawing showing the forces acting on the loads. 330 C2.56. From a height H above the ground, a steel ball begins to fall freely, which after a time t = 0.4 c collides with a plate inclined at an angle of 30 to the horizon. After an absolutely elastic impact, it moves along a trajectory, the top point of which is at a height h = 1.4 m above the ground. What is the height H? Make a schematic drawing explaining the solution. H = 2 m. C2.57. The photograph shows a setup for studying the uniform motion of a bar 1 with a mass of 0.1 kg, on which there is a load 2 with a mass of 0.1 kg. What is the work of the traction force when moving the bar with a load on the surface of the table for a distance of 15 cm? Write your answer to the nearest hundredth. 0.06 J

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in the tasks of the exam

The ball is thrown vertically upwards. The figure shows a graph of the change in the kinetic energy of the ball as it rises above the point of throw. What is the potential energy of the ball at a height of 2 m? Solution:

The figure shows a graph of the change over time in the kinetic energy of a child swinging on a swing. At the moment corresponding to point A on the graph, its potential energy, counted from the equilibrium position of the swing, is 1) 10 J 2) 20J 3) 30 J 4) 25 J

A small puck of mass 2 g can slide without friction along a cylindrical recess with a radius of 0.5 m. Starting from above, it collides with another similar puck resting below. What is the amount of heat released as a result of the inelastic collision of the washers?

Solution:

A weight suspended on a thread performs harmonic oscillations. The table shows the coordinates of the weight at regular intervals. What is the maximum speed of the weight?

The ball slides without friction from the upper end of the inclined chute, turning into a "dead loop" with radius R. What is the pressure force of the ball on the chute at the top point of the loop, if the mass of the ball 0.1 kg, and the upper end of the gutter is raised to a height h=3R in relation to the lower point of the "dead loop"?

A small puck after a push gains speed υ = 2 m/s and slides along the inner surface of a smooth fixed ring with a radius R= 0.14 m. At what height h Does the puck come off the ring and start to fall freely?

A ball of mass 0.2 kg on a thread 0.9 m long is swung so that each time the ball passes the equilibrium position, a force of 0.1 N is applied to it for a short time interval of 0.01 s, directed parallel to the speed. After how many complete oscillations will the ball on the string deviate through an angle of 60°?

A ball floats up from the bottom of the aquarium and jumps out of the water. In air, it has kinetic energy, which it acquired by reducing: 1) the internal energy of water 2) potential energy of the ball 3) potential energy of water 4) kinetic energy of water

The parachutist descends at a constant speed. What energy transformations take place?

• The potential energy of a parachutist is completely converted into his kinetic energy.

• The kinetic energy of the skydiver is completely converted into its potential energy.

• The kinetic energy of the skydiver is completely converted into the internal energy of the skydiver and the air

• The energy of the interaction of a skydiver with the Earth is converted into the internal energy of the interacting bodies due to the forces of air resistance

In a thermally insulated vessel, 1 mol of hydrogen with an average molecular kinetic energy of 1 10-20 J and 4 mol of oxygen with an average molecular kinetic energy of 2 10-20 J are mixed. What is the average kinetic energy of the molecules after mixing?

I law of thermodynamics

The first law of thermodynamics is written as follows: Q=A+ΔU, where Q- the amount of heat received by the gas, A - the work done by the gas. During the process carried out with the gas, its internal energy decreased, while the gas was compressed. What are the signs Q And A?

What amount of heat must be transferred to 1 mole of a monatomic gas in order to double its volume in an isobaric process, if the initial temperature of the gas is T?

An ideal monatomic gas is located in a vessel with rigid walls with a volume of 0.6 m3. When heated, its pressure increased by 3 kPa. How much has the internal energy of the gas increased?

The graph shows the process of changing the state of the gas. The gas gives off 50 kJ of heat. What is the work done by external forces?

A monatomic ideal gas goes through the cyclic process shown in the figure. The mass of the gas is constant. For a cycle from the heater, the gas receives the amount of heat Qн = 8 kJ. What is the work done by the gas per cycle?

A horizontal cylinder is fixed in a vacuum. The cylinder contains 0.1 mole of helium, locked by a piston. A piston of mass 90 g is held by stops and can slide along the walls of the cylinder without friction. A bullet of mass 10 g, flying horizontally at a speed of 400 m/s, hits the piston and gets stuck in it. How will the helium temperature change when the piston stops in the leftmost position? Assume that during the movement of the piston, the gas does not have time to exchange heat with the vessel and piston.

A horizontally located positively charged plate creates a vertically directed uniform electric field with a strength of E=105 V/m. A ball with a mass m=40 g falls onto it from a height h=10 cm, having a negative charge q=-10-6 C and an initial velocity v0=2m/s directed vertically downwards. What energy will the ball transfer to the plate in a perfectly inelastic impact?

If we move apart the plates of a capacitor connected to the terminals of a galvanic cell, then its energy:

• Decreases because the distance between positive and negative charges on the plates increases

• Increases because force pushing the plates apart does work

• Decreases, because with a constant potential difference between the plates, the capacitance of the capacitor decreases

• Increases, because with a constant charge on the plates of the capacitor, its capacitance decreases

Two capacitors with capacities of 4 uF and 8 uF are charged to a voltage of 3 V each, and then the “plus” of one of them is connected to the “minus” of the other and the free terminals are connected with a 1000 Ohm resistor. How much heat will be released in the resistor?

A DC motor is connected to a current source and lifts a load of 1 g at a speed of 4 cm/s. The voltage at the motor terminals is 4 V, the current is 1 mA. How much heat will be released in the motor winding in 5 s?

The voltage at the capacitor terminals in the oscillatory circuit changes over time according to the graph in the figure. What energy transformation takes place in the circuit in the interval from 2⋅10-3 s to 3⋅10-3 s?

• 1) the energy of the magnetic field of the coil decreases from the maximum value to 0

• 2) the energy of the magnetic field of the coil is converted into the energy of the electric field of the capacitor

• 3) the energy of the electric field of the capacitor increases from 0 to the maximum value

• 4) the energy of the electric field of the capacitor is converted into the energy of the magnetic field of the coil.

The capacitance of the capacitor included in the AC circuit is 6 uF. The equation for voltage fluctuations on the capacitor is: U=50 cos(1000t), where all quantities are expressed in SI. Find the amplitude of the current

At what voltage on the current source (see figure) will the electrons knocked out of one plate not reach the second? Incident light wavelength λ= 663 nm, work function A= 1.5 eV.

A free pion (π0-meson) with a rest energy of 135 MeV moves at a speed V which is much less than the speed of light. As a result of its decay, two γ-quanta were formed, one of them propagating in the direction of pion motion, and the other in the opposite direction. The energy of one quantum is 10% more than the other. What is the speed of the pion before decay?

Themes USE codifier Keywords: work of force, power, kinetic energy, potential energy, law of conservation of mechanical energy.

We begin the study of energy - the fundamental physical concept. But first you need to deal with the other physical quantity- the work of force.

Job.

Let a constant force act on the body and the body, moving in a straight line on a horizontal surface, has made a displacement . Force is not necessarily the direct cause of movement (thus, gravity is not the direct cause of movement of a cupboard that is being moved around a room).

Let us first assume that the vectors of force and displacement are co-directed (Fig. 1; other forces acting on the body are not indicated)

Rice. 1.A=Fs

In this simplest case, work is defined as the product of the modulus of force and the modulus of displacement:

. (1)

The unit of work is the joule (J): J = N m. Thus, if under the action of a force of 1 N the body moves 1 m, then the force does work of 1 J.

The work of a force perpendicular to the displacement is, by definition, considered to be zero. So, in this case, the force of gravity and the reaction force of the support do not do work.

Now let the force vector form with the displacement vector sharp corner(Fig. 2).

Rice. 2.A=Fscos

We decompose the force into two components: (parallel to the displacement) and (perpendicular to the displacement). Only does the work. Therefore, for the work of the force we get:

. (2)

If the force vector forms an obtuse angle with the displacement vector, then the work is still determined by formula (2) . In this case, the work is negative.

For example, the work of the sliding friction force acting on the body in the situations considered will be negative, since the friction force is directed opposite to the displacement. In this case we have:

And for the work of the friction force we get:

where is the mass of the body, is the coefficient of friction between the body and the support.

Relation (2) means that the work is the scalar product of the force and displacement vectors:

This allows you to calculate the work through the coordinates of the given vectors:

Let several forces act on the body and be the resultant of these forces. For the work of the force we have:

where is the work of forces. So, the work of the resultant forces applied to the body is equal to the sum of the work of each force separately.

Power.

Often the speed with which the work is done is important. Say, in practice, it is important to know what work a given device can do in a fixed time.

Power is a measure of the rate at which work is done. Power is the ratio of work to time for which this work is done:

Power is measured in watts (W). 1 W \u003d 1 J / s, that is, 1 W is such a power at which work of 1 J is done in 1 s.

Suppose that the forces acting on the body are balanced, and the body moves uniformly and in a straight line with a speed. In this case, there is a useful formula for the power developed by one of the acting forces.

In time, the body will move. The work done by the force will be:

From here we get the power:

where is the angle between the force and velocity vectors.

Most often, this formula is used in a situation where - the "traction" force of the car engine (which is actually the friction force of the driving wheels on the road). In this case , and we get simply:

mechanical energy.

Energy is a measure of the movement and interaction of any objects in nature. There are various forms of energy: mechanical, thermal, electromagnetic, nuclear. . .

Experience shows that energy does not appear from nowhere and does not disappear without a trace, it only passes from one form to another. This is the most general wording. law of conservation of energy.

Each type of energy is some mathematical expression. The law of conservation of energy means that in every natural phenomenon a certain sum of such expressions remains constant over time.

Energy is measured in joules, just like work.

mechanical energy is a measure of the movement and interaction of mechanical objects (material points, solid bodies).

The measure of body movement is kinetic energy. It depends on the speed of the body. The measure of the interaction of bodies is potential energy. It depends on relative position tel.

The mechanical energy of a system of bodies is equal to the sum of the kinetic energy of the bodies and the potential energy of their interaction with each other.

Kinetic energy.

The kinetic energy of the body (taken as material point) is called the quantity

where is the mass of the body and is its speed.

The kinetic energy of a system of bodies is the sum of the kinetic energies of each body:

If a body moves under the action of a force, then the kinetic energy of the body, generally speaking, changes with time. It turns out that the change in the kinetic energy of the body over a certain period of time is equal to the work of the force. Let us show this for the case of rectilinear uniformly accelerated motion.

Let be the initial speed and be the final speed of the body. Let's choose an axis along the trajectory of the body (and, accordingly, along the force vector ). For the work of the force we get:

(we used the formula for derived in the article " Uniformly accelerated motion"). Note now that in this case the velocity projection differs from the velocity modulus only in sign; therefore and . As a result, we have:

which is what was required.

In fact, the relation is also valid in the most general case of curvilinear motion under the action of a variable force.

Theorem on kinetic energy. The change in the kinetic energy of the body is equal to the work done by the external forces applied to the body during the considered period of time.

If the work of external forces is positive, then the kinetic energy increases ( class="tex" alt="\Delta K>0">, тело разгоняется).!}

If the work of external forces is negative, then the kinetic energy decreases (the body slows down). An example is braking under the action of a friction force, the work of which is negative.

If the work of external forces is equal to zero, then the kinetic energy of the body does not change during this time. Non-trivial example - uniform motion in a circle, performed by a load on a thread in a horizontal plane. The force of gravity, the reaction force of the support, and the force of the thread tension are always perpendicular to the speed, and the work of each of these forces is zero for any period of time. Accordingly, the kinetic energy of the load (and hence its speed) remains constant during the movement.

Task. A car is moving along a horizontal road at a speed and starts to brake sharply. Find the distance traveled by the car to a complete stop, if the coefficient of friction of the tires on the road is .

Solution. Initial kinetic energy of the car, final kinetic energy. Change in kinetic energy.

The force acting on the car is gravity, the reaction of the support, and the force of friction. The force of gravity and the reaction of the support, being perpendicular to the movement of the car, do no work. The work of the friction force:

From the kinetic energy theorem we now obtain:

Potential energy of a body near the surface of the Earth.

Consider a body of mass , located at a certain height above the Earth's surface. We consider the height to be much less than the earth's radius. We neglect the change in the force of gravity in the process of moving the body.

If the body is at a height, then the potential energy of the body is by definition equal to:

where is the free fall acceleration near the Earth's surface.

Height does not have to be measured from the surface of the earth. As we will see below (formulas (3) , (4) ), physical meaning possesses not the potential energy itself, but its change. And the change in potential energy does not depend on the reference level. The choice of the zero level of potential energy in a particular problem is dictated solely by considerations of convenience.

Find the work done by gravity when moving the body. Suppose that the body moves in a straight line from a point at a height to a point at a height (Fig. 3).

Rice. 3.A=mg(h1-h2)

The angle between the force of gravity and the displacement of the body will be denoted by . For the work of gravity we get:

But, as can be seen from Fig. 3 , . That's why

. (3)

Considering that , we also have:

. (4)

It can be proved that formulas (3) and (4) are valid for any trajectory along which the body moves from point to point , and not only for a straight line segment.

The work of gravity does not depend on the shape of the trajectory along which the body moves, and is equal to the difference in the values ​​of potential energy at the initial and final points of the trajectory. In other words, the work of gravity is always equal to the change in potential energy with the opposite sign. In particular, the work done by gravity along any closed path is zero.

The force is called conservative , if the work of this force when moving the body does not depend on the shape of the trajectory, but is determined only by the initial and final position of the body. Gravity is thus conservative. The work of a conservative force along any closed path is zero. Only in the case of a conservative force is it possible to introduce such a quantity as potential energy.

Potential energy of a deformed spring.

Consider a stiffness spring. The initial deformation of the spring is . Suppose
that the spring is deformed to some finite amount of deformation . What is the work done by the elastic force of the spring?

In this case, you cannot multiply the force by displacement, since the elastic force changes during the deformation of the spring. To find the work of a variable force, integration is required. We will not present the derivation here, but immediately write out the final result.

It turns out that the spring force is also conservative. Its work depends only on the quantities and and is determined by the formula:

Value

is called the potential energy of the deformed spring (x is the amount of deformation).

Hence,

which is completely similar to formulas (3) and (4) .

The law of conservation of mechanical energy.

Conservative forces are called so because they conserve the mechanical energy of a closed system of bodies.

The mechanical energy of a body is equal to the sum of its kinetic and potential energies:

The mechanical energy of a system of bodies is equal to the sum of their kinetic energies and the potential energy of their interaction with each other.

Let us assume that the body moves under the action of gravity and/or spring force. We will assume that there is no friction. Let the kinetic and potential energies of the body be equal in the initial position and , in the final position - and . We denote the work of external forces when moving the body from the initial position to the final one.

According to the kinetic energy theorem

But the work of conservative forces is equal to the difference in potential energies:

From here we get:

The left and right parts of this equation represent the mechanical energy of the body in the initial and final positions:

Consequently, when a body moves in a gravitational field and/or on a spring, the mechanical energy of the body remains unchanged in the absence of friction. A more general assertion is also true.

Law of conservation of mechanical energy . If only conservative forces act in a closed system, then the mechanical energy of the system is conserved.

Under these conditions, only energy transformations can occur: from kinetic to potential and vice versa. The total supply of mechanical energy of the system remains constant.

The law of change of mechanical energy.

If there are resistance forces between the bodies of a closed system (dry or viscous friction), then the mechanical energy of the system will decrease. So, the car stops as a result of braking, the oscillations of the pendulum gradually die out, etc. The friction forces are non-conservative: the work of the friction force obviously depends on the path along which the body moves between these points. In particular, the work of the friction force along a closed path is not equal to zero.

Consider again the motion of a body in a gravitational field and/or on a spring. In addition, a friction force acts on the body, which, over the considered period of time, performs negative work. The work of conservative forces (gravity and elasticity) is still denoted by .

The change in the kinetic energy of the body is equal to the work of all external forces:

But, therefore

On the left side is the value - the change in the mechanical energy of the body:

So, when a body moves in a gravitational field and/or on a spring, the change in the mechanical energy of the body is equal to the work of the friction force. Since the work of the friction force is negative, the change in mechanical energy is also negative: the mechanical energy decreases.
A more general assertion is also true.

The law of change of mechanical energy. The change in the mechanical energy of a closed system is equal to the work of the friction forces acting inside the system.

It is clear that the law of conservation of mechanical energy is a special case of this statement.

Of course, the loss of mechanical energy does not contradict the general physical law of conservation of energy. In this case, the mechanical energy is converted into the energy of the thermal motion of the particles of matter and their potential energy of interaction with each other, i.e., it is converted into the internal energy of the bodies of the system.