Themes of the USE codifier: work of force, power, kinetic energy, potential energy, the law of conservation of mechanical energy.
We begin to study energy - a fundamental physical concept. But first you need to deal with another physical quantity - the work of force.
Job.
Let a constant force act on the body and the body, moving rectilinearly along the horizontal surface, made a displacement. Force is not necessarily the direct cause of movement (for example, gravity is not the direct cause of movement of a cabinet that is being moved around the room).
First, assume that the vectors of force and displacement are codirectional (Fig. 1; other forces acting on the body are not indicated)
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| Rice. 1.A = Fs |
In this simplest case, work is defined as the product of the force modulus by the displacement modulus:
. (1)
The unit of measure of work is the joule (J): J = N m.Thus, if under the action of a force of 1 N the body moves by 1 m, then the force performs work of 1 J.
The work of the 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 the work.
Now let the force vector form an acute angle with the displacement vector (Fig. 2).
 |
| Rice. 2.A = Fs cos |
Let us decompose the force into two components: (parallel to displacement) and (perpendicular to displacement). He does the work only. 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 turns out to be 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 body mass, is the coefficient of friction between the body and the support.
Relation (2) means that the work is the dot product of the force and displacement vectors:
This allows you to compute 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 are 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.
The speed with which the work is done often matters. For example, in practice, it is important to know what work a given device can perform in a fixed time.
Power - This is a value that characterizes the speed of work. Power is the ratio of work to the time during which this work is completed:
Power is measured in watts (W). 1 W = 1 J / s, that is, 1 W is the 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 rectilinearly with speed. In this case, there is a useful formula for the power developed by one of the acting forces.
During the time the body will move. The work of force will be equal to:
From here we get the power:
where is the angle between the vectors of force and velocity.
Most often this formula is used in a situation when - 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 out of nowhere and does not disappear without a trace, it only passes from one form to another. This is the most general formulation energy conservation law.
Each type of energy represents some kind of mathematical expression. The law of conservation of energy means that in every natural phenomenon, a certain amount 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, solids).
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 the relative position of the bodies.
The mechanical energy of a system of bodies is equal to the sum of the kinetic energy of bodies and the potential energy of their interaction with each other.
Kinetic energy.
The kinetic energy of a body (taken as a material point) is the quantity
where is the mass of the body, is its speed.
The kinetic energy of a system of bodies is the sum of the kinetic energies of each body:
If the 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 - initial speed, - 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 "Equally accelerated motion"). Note now that in this case the velocity projection differs from the velocity modulus only by the sign; that's why . As a result, we have:
as required.
In fact, the relationship is also valid in the most general case of curvilinear motion under the action of an alternating force.
Kinetic energy theorem. The change in the kinetic energy of the body is equal to the work done by 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 = "(! LANG: \ 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 influence of a frictional 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. A non-trivial example is a uniform movement along a circle performed by a load on a thread in a horizontal plane. The force of gravity, the force of reaction of the support and the force of tension on the thread are always perpendicular to the speed, and the work of each of these forces is equal to zero for any period of time. Accordingly, the kinetic energy of the load (and hence its speed) remains constant during the movement.
Task. The car drives on 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 vehicle, final kinetic energy. Change in kinetic energy.
The vehicle is affected by gravity, bearing reaction and friction. The force of gravity and the reaction of the support, being perpendicular to the movement of the car, do not perform work. Frictional force work:
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 acceleration of gravity near the Earth's surface.
Altitude does not have to be measured from the surface of the Earth. As we will see below (formulas (3), (4)), it is not the potential energy itself that has physical meaning, but its change. And the change in potential energy does not depend on the counting level. The choice of the zero level of potential energy in a specific problem is dictated solely by considerations of convenience.
Let's find the work done by gravity when the body is moving. Suppose that the body moves along a straight line from a point at a height to a point at a height (Fig. 3).
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| Rice. 3.A = mg (h1-h2) |
The angle between the force of gravity and the displacement of the body will be denoted. 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 the force 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 the 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 of gravity along any closed path is zero.
The power is called conservative if, when the body moves, the work of this force 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 the conservative force on 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 the deformed spring.
Consider a spring of stiffness. The initial deformation of the spring is equal to. Suppose
that the spring is deformed to some finite amount of deformation. What is the work of the spring force equal to?
In this case, the force per movement cannot be multiplied, since the elastic force changes during the deformation of the spring. Integration is required to find the work of variable force. We will not present the conclusion here, but will immediately write out the final result.
It turns out that the spring force is also conservative. Its work depends only on the quantities and is determined by the formula:
The quantity
is called the potential energy of the deformed spring (x is the amount of deformation).
Hence,
which is completely analogous to formulas (3) and (4).
The law of conservation of mechanical energy.
Conservative forces are called so because they preserve 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.
Suppose that the body moves under the action of gravity and / or the spring force. We will assume that there is no friction. Let in the initial position the kinetic and potential energies of the body are equal and, in the final position - and. The work of external forces when the body moves from the initial position to the final one will be denoted.
By 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 sides of this equality represent the mechanical energy of the body in the initial and final position:
Consequently, when a body moves in a gravity field and / or on a spring, the body's mechanical energy remains unchanged in the absence of friction. A more general statement is also valid.
Mechanical energy conservation law ... If only conservative forces act in a closed system, then the mechanical energy of the system is conserved.
Under these conditions, only transformations of energy can occur: from kinetic to potential and vice versa. The total supply of mechanical energy in the system remains constant.
The law of change in 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. Thus, the car stops as a result of braking, the oscillations of the pendulum gradually damp, etc. 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 zero.
Consider again the motion of a body in a gravity field and / or on a spring. In addition, a frictional force acts on the body, which performs negative work during the considered period of time. We continue to denote the work of conservative forces (gravity and elasticity).
The change in the kinetic energy of the body is equal to the work of all external forces:
But therefore
On the left side there is a value - a change in the mechanical energy of the body:
So, when a body moves in a gravity field and / or on a spring, the change in the body's mechanical energy 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: mechanical energy decreases.
A more general statement is also valid.
The law of change in 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 the substance and their potential energy of interaction with each other, that is, it is converted into the internal energy of the bodies of the system.
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1 C1.1. After the jolt, the piece of ice rolled into a hole with smooth walls, in which it can move practically without friction. The figure shows a graph of the dependence of the interaction energy of a piece of ice with the Earth on its coordinates in the pit. At some point in time, the piece of ice 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 piece of ice slip out of the hole? Explain the answer, indicating what physical laws you used to explain. C1.2. After the jolt, the piece of ice rolled into a hole with smooth walls, in which it can move practically without friction. The figure shows a graph of the dependence of the interaction energy of a piece of ice with the Earth on its coordinate in the pit. At some point in time, the piece of ice 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 piece of ice slip out of the hole? Explain the answer, indicating what physical laws you used to explain. C2.1. C2.2. With F781, a Body weighing 1 kg was thrown from the Earth's surface at a speed of 20 m / s at an angle of 45 0 to the horizon. What work has been done by gravity during the flight of the body (from throwing to falling to the ground)? Neglect air resistance. 0 C2.4. C38106 A sled with riders with a total mass of 100 kg descends from a mountain 8 m high and 100 m long. What is the average force of resistance to the movement of the sled, if at the end of the mountain they reach a speed of 10 m / s, and the initial speed is zero? 30 H C2.5. A bar with a mass of m 1 = 600 g, moving at a speed of v 1 = 2 m / s, collides with a stationary bar with a mass of m 2 = 200 g. What will be the speed of the first bar after the collision? Consider the impact as central and absolutely elastic. 1 m / s. C2.6. A bar with a mass of m 1 = 500 g slides along an inclined plane from a height h and, moving along a horizontal surface, collides with a stationary bar with a mass of m 2 = 300 g. As a result of an absolutely inelastic collision, the total kinetic energy of the bars becomes 2.5 J. Determine the height inclined plane h. Neglect friction when driving. Consider that the inclined plane smoothly turns into a horizontal one. h = 0.8 m. C2.7. A bar with a mass of m 1 = 500 g slides along an inclined plane with a height of h = 0.8 m and collides with a stationary bar with a mass of m 2 = 300 g, lying on a horizontal surface. Considering the collision elastic, determine the kinetic energy of the first bar after the collision. Neglect friction when driving.
2 Answer 0.25 J. C2.8. On a smooth horizontal plane there is a smooth slide with a height of H = 24 cm and a mass of M = 1 kg, and on its top there is a small washer with a mass of m = 200 g (see figure). After a slight push, the washer slides off the hill and moves perpendicular to the wall, fixed in a vertical position on a plane. At what speed does the puck approach the wall along the plane? C2.9. A puck thrown along an inclined plane slides along it, moving up and then moving down. The plot of the puck speed module versus time is shown 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 with a mass of m 1 = 500 g slides along an inclined plane from a height of h = 0.8 m and, moving along a horizontal surface, collides with a stationary bar with a mass of m 2 = 300 g. Considering the collision to be absolutely inelastic, determine the total kinetic energy of the bars after the collision. Neglect friction when driving. Consider that the inclined plane smoothly turns into a horizontal one. E k = 2.5 J. C2.11. A bar with a mass of m 1 = 500 g slides along an inclined plane with a height of h = 0.8 m and collides with a stationary bar with a mass of m 2 = 300 g lying on a horizontal surface. Considering the collision elastic, determine the kinetic energy of the first bar after the collision. Neglect friction when driving. 0.25 J C2.12. A bar with a mass of m 1 = 0.5 kg slides along an inclined plane from a height of h = 0.8 m and, moving along a horizontal surface, collides with a stationary bar with a mass of m 2 = 0.3 kg. Assuming the collision is completely inelastic, calculate the total kinetic energy of the bars after the collision. Neglect friction when driving. Consider that the inclined plane smoothly turns into a horizontal one. S2.13. A bar with a mass of m 1 = 600 g, moving at a speed of v 1 = 2 m / s, collides with a stationary bar with a mass of m 2 = 200 g. What will be the speed of the first bar after the collision? Consider the impact as central and absolutely elastic. 1 m / s
3 C2.14. A bar of mass m slides along the horizontal surface of the table and catches up with a bar 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 to the moment when their speed becomes 2v o / 7? 0.5 m C2.15. The washer of mass m starts to move along the groove AB from point A from the 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 due to friction decreases by ΔE = 2 J. At point B, the washer flies out of the chute at an angle of α = 15 to the horizon and falls to the ground at point D, located on the same horizontal line with point B (see figure). BD = 4 m. Find the mass of the washer t. Neglect the air resistance. t = 0.1 kg. S2.16. A washer with a mass of m = 100 g begins to move along the groove 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 movement along the chute, the mechanical energy of the washer due to friction decreases by ΔE = 2 J. ground at point D. located on the same horizontal with point B (see figure). Find BD. Neglect air resistance. BD = 4 m C2.17. A washer with a mass of m = 100 g begins to move along the groove 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 due to friction decreases by ΔE. At point B, the washer flies out of the gutter at an angle of α = 15 to the horizontal and falls to the ground at point D, which is on the same horizontal as point B (see figure). BD = 4 m. Find the value of ΔE. Neglect air resistance. ΔE = 2 J. C2.18. CE1284 A slide with two peaks, the heights of which are h and 3h, rests on a smooth horizontal table surface (see figure). On the right top of the slide there is a washer, the mass of which is 12 times less than the mass of the slide. From an insignificant push, the puck and the slide start to move, and the puck moves to the left, not 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 is on the left top of the slide.
4 C2.19. After impact, the small puck slides up the inclined plane from point A (see figure). At point B, the inclined plane without kink passes into the outer surface of the 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 off 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. After pushing, a small washer acquires a velocity v = 2 m / s and slides along the inner surface of a smooth fixed ring with a radius of R = 0.14 m. At what height h does the washer break off from the ring and begin to fall freely? h 0.18m. S2.21. A piece of plasticine collides with a bar resting on the horizontal surface of the table and sticks to it. The speed of the plasticine before the impact is v pl = 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 the stuck block with plasticine move to 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 the plasticine and the bar before the impact are oppositely directed 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 stuck block with plasticine move to 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 the plasticine and the bar before the 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 stuck block with plasticine move to 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 the plasticine and the bar before the 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 time when the speed of the sticking bar and plasticine has decreased by 2 times, they have moved by 0.22 m. Determine the coefficient of friction μ of the bar on the table surface. μ = 0.17. S2.25. A trolley weighing 0.8 kg moves by inertia at a speed of 2.5 m / s. A piece of plasticine weighing 0.2 kg vertically falls onto the cart from a height of 50 cm and sticks to it. Calculate the energy that has passed into the internal with this impact. Q = 1.5 J.
5 C2.26. The bullet flies horizontally with a speed of v 0 = 150 m / s, breaks through a block standing on the horizontal surface of the ice 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 the ice is μ = 0.1. How far S will the bar move to the moment when its speed decreases by 10%? S2.27. A bullet flying horizontally at a speed of v o = 120 m / s, pierces the box lying on the horizontal surface of the table and continues to move 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 to the moment when its speed is halved? S2.28. A pile weighing 150 kg is immersed in the ground by 10 cm from the impact of a pile driver weighing 450 kg, falling freely from a height of 5 m. Determine the resistance force of the soil, considering it constant, and the impact is absolutely inelastic. Disregard the change in the potential energy of the pile in the gravitational field of the Earth. S2.29. The cannon, fixed at a height of 5 m, shoots 10 kg shells in the horizontal direction. Due to the recoil, its barrel, which has a mass of 1000 kg, compresses the spring of stiffness N / m by 1 m, which reloads the gun. Assuming that the relative fraction η = 1/6 of the recoil energy goes to compress the spring, find the range of the projectile. S2.30. A spring pistol was fired vertically downward at a target located at a distance of 2 m from it. Having completed the work of 0.12 J, the bullet got stuck in the target. What is the mass of a bullet if the spring was compressed by 2 cm before firing, and its stiffness was 100 N / m? S2.31. A massive weight is attached to one end of a light spring with rigidity k = 100 N / m, lying on a horizontal plane, the other end of the spring is fixed motionlessly (see figure). The coefficient of friction of the load along the plane is μ = 0.2. The load is displaced horizontally by stretching the spring, then released at an initial velocity of zero. The load moves in one direction and then stops in a position in which the spring is already compressed. The maximum tension of the spring, at which the load moves in this way, is equal to d = 15 cm. Find the mass m of the load. S2.32. 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 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 regu with a speed of 0.3 m / s, and if he pulled only the second at a speed of 0.4 m / s. How fast will the boat approach the shore when it is being pulled by both fishermen? Disregard water resistance. 0.5 m / s. S2.33. What is the average pressure of the powder gases in the barrel of a gun if the velocity of the projectile ejected from it is 1.5 km / s? The barrel length is 3 m, its diameter is 45 mm, the mass of the projectile is 2 kg. (Friction is negligible.) P = 4, Pa. S2.34. When performing the "Flying Cyclist" trick, the rider moves along the springboard under the influence of gravity, starting from a state of rest from a height H (see figure). At the edge of the springboard, the rider's speed is directed at such an angle to the horizon that the range of his flight is maximum. Flying through the air, the rider lands on a horizontal table at the same height as the edge of the springboard. What is the flight altitude h on this springboard? Neglect air resistance and friction. lifting height C2.35. When performing the "Flying Cyclist" trick, the rider moves along the springboard under the influence of gravity, starting from a state of rest from 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. Flying through the air, the rider lands on a horizontal table at the same height as the edge of the springboard. What is the flight range L on this springboard? Neglect air resistance and friction. flight range C2.36. When performing the "Flying Cyclist" trick, the rider moves along a smooth springboard under the influence of gravity, starting from a state of rest from 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 muzzle velocity of a projectile fired vertically upward from the cannon is 500 m / s. At the point of maximum rise, the projectile exploded into two fragments. The first fell to the ground near the point of the shot, having a speed 2 times the initial velocity of the projectile, and the second in the same place - 100 s after the burst. What is the ratio of the mass of the first fragment to the mass of the second fragment? Neglect air resistance.
7 C2.38. A projectile weighing 4 kg, flying at a speed of 400 m / s, is torn into two equal parts, one of which flies in the direction of movement 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. The speed of the fragment flying in the direction of movement of the projectile is 900 m / s. Find ΔE. ΔE = 0.5 MJ. S2.39. A projectile weighing 4 kg, flying at a speed of 400 m / s, is torn into two equal parts, one of which flies in the direction of movement 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 splinter flying in the direction of travel of the projectile. v 1 = 900 m / s. S2.40. In flight, the projectile breaks into two equal parts, one of which continues to move in the direction of the projectile movement, 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 the value of ΔE. The velocity module of the fragment moving in the direction of movement of the projectile is V 1, and the velocity module of the second fragment is V 2. Find the mass of the projectile. S2.41. Two bodies, the masses of which, respectively, m 1 = 1 kg and m 2 = 2 kg, slide on a smooth horizontal table (see figure). The speed of the first body is v 1 = 3 m / s, the speed of the second body is v 2 = 6 m / s. How much heat will be released when they collide and move on, grappling together? There is no rotation in the system. Disregard the action of external forces. Q = 15 (J). S2.43. A projectile with a mass of 2 tons, moving at a speed of v 0, is torn into two equal parts, one of which continues to move in the direction of movement of the projectile, and the other in the opposite direction. At the moment of rupture, the total kinetic energy of the fragments is uvev 2 90 m 2 v 1 m 1 C2.42. The figure shows a photograph of the installation for studying the sliding of the carriage (1) weighing 40 g along an inclined plane at an angle of 30. At the moment of the start of movement, the upper sensor (2) turns on the stopwatch (3). When the carriage passes the lower sensor (4), the stopwatch turns off. Estimate the amount of heat released when the carriage slides along an inclined plane between the sensors Q 0.03 (J). 3
8 increases due to the energy of the explosion by the value of ΔE. The speed of the fragment moving in the direction of movement of the projectile is equal to v 1. Find ΔE. S2.44. A pendulum thread with a length of l = 1 m, to which a load weighing 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 tensile force modulus of the thread at the moment the pendulum passes the equilibrium position T = 2 N. What is the angle α? S2.45. An elastic ball moving along a smooth horizontal plane with speed experiences an absolutely elastic non-sided 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 after the collision? S2.46. A small ball is suspended on an inextensible and weightless thread l = 0.5 m long. The ball in the equilibrium position is reported with a horizontal velocity υ 0 = 4 m / s. Calculate the maximum height h, starting from the equilibrium position of the ball, after which the ball will stop moving around a circle of radius l. 0.7 m. C2.47. Two balls, the masses of which differ by 3 times, hang in contact on vertical threads (see figure). The light ball is deflected at an angle of 90 and released without initial velocity. Find the ratio of the momentum of a light ball to the momentum of a heavy ball immediately after an absolutely elastic central collision. S2.48. 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. How high will the balls rise after impact, if this impact is absolutely inelastic? h = 0.05 m. C2.49. Two balls, the masses of which differ by 3 times, hang, touching, on vertical threads (see figure). The light ball is deflected at 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 absolutely elastic central impact? S2.50. A ball weighing 1 kg, suspended on a thread 90 cm long, is retracted from the equilibrium position at an angle of 60 and released. At the moment the ball passes the equilibrium position at.
9 he is hit by a bullet weighing 10 g, flying towards the ball at a speed of 300 m / s. It punches 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 considered unchanged, the diameter of the ball is negligible in comparison with the length of the thread.) C2.51. A ball weighing 1 kg, suspended on a thread 90 cm long, is retracted from the equilibrium position at an angle of 60 ° and released. At the moment the ball passes the equilibrium position, a bullet with a mass of 10 g hits it, flying towards the ball. She punches him and continues to move horizontally. Determine the change in the speed of the bullet as a result of hitting the ball if, while continuing to move in the same direction, it deflects at 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.) 100 m / s. S2.52. A ball weighing 1 kg, suspended on a thread 90 cm long, is retracted from the equilibrium position at 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. At what speed did the bullet fly if the ball, continuing to move in the horizontal direction, deflects at 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. S2.53. The figure shows a vertically positioned spring pendulum 2. The mass of the pendulum platform is m 2 = 0.2 kg, the length of the spring is L = 10 cm. A washer 1 with a mass of m 1 = 0.1 kg falls onto the spring pendulum from a height of H = 25 cm. After the collision, the platform with the washer vibrates as a whole. Calculate the energy that has passed into internal energy when the puck hits the pendulum platform. 0.1 J C2.54. The 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. Weight m is located at a point at the top of the sphere (see figure). In the course of the resulting movement, the load m breaks away from the surface of the sphere, passing an arc 30 along it. Find the mass M if m = 100 g. The dimensions of the load m are negligible in comparison with the radius of the sphere. Friction is neglected. Make a schematic drawing showing the forces acting on the weights.
10 C2.55. The 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. Weight m is located at a point at the top of the sphere (see figure). In the course of the resulting movement, the load m breaks away from the surface of the sphere, passing an arc 30 along it. Find the mass M if m = 100 g. The dimensions of the load m are negligible in comparison with the radius of the sphere. Friction is neglected. Make a schematic drawing showing the forces acting on the weights. 330 g C2.56. From a height H above the ground, a steel ball begins to fall freely, which after a time t = 0.4 s collides with a plate inclined at an angle of 30 to the horizon. After an absolutely elastic impact, it moves along a trajectory, the upper point of which is at a height of h = 1.4 m above the ground. What is the height H? Make a schematic drawing to illustrate the solution. H = 2 m. C2.57. The photograph shows an installation for studying the uniform motion of a bar 1 weighing 0.1 kg, on which there is a load 2 weighing 0.1 kg. What is the work of the traction force when moving a bar with a load over the table surface at a distance of 15 cm? Write down your answer to the nearest hundredths. 0.06 J
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Lesson "Solving USE problems on the topic" Conservation laws in mechanics "
Target: formation of skills in solving problems on this topic
Tasks:
recall the theory on the topic "Law of conservation of momentum", "Law of conservation of energy"
be able to apply laws to solving the problems of the exam on these topics
learn how to apply conservation laws to solving more complex problems
During the classes:
Organizing time
The teacher formulates the condition of the problem of part C, directs the students to the solution of this problem. Asks what knowledge may be needed to solve a problem of this type.
Problem C2, 2009
Two balls, the masses of which differ by 3 times, hang in contact on vertical threads. The light ball is deflected at an angle of 90 ° and released without initial velocity. Find the ratio of the momentum of a light ball to the momentum of a heavy ball immediately after an absolutely elastic central collision.
How to immediately guess that in this problem it is necessary to use the laws of conservation of momentum and energy, and not try to solve it "ordinary"
in a way, that is, making a drawing with all forces acting on the body and then applying Newton's laws?
– This problem considers uneven curvilinear traffic
body, and the resultant forces applied to the body changes over time.
Students are presented with multiple choice problems.
1. The figure shows a load suspended on a thread and performing free vibrations like a pendulum. Within what limits does its potential energy change during these fluctuations of the load?
The total mechanical energy of the load at the moment of deviation from the equilibrium position is 10 J.
A) Potential energy does not change and is equal to 10 J;
B) Potential energy does not change and is equal to 5 J;
V) Potential energy varies from 0 to 10 J;
D) Potential energy varies from 0 to 5 J.
Answer: 3
3. The ball hit the wall, and the speed of the ball immediately after the hit is half its speed immediately before the hit. What is the kinetic energy of the ball before the impact if the amount of heat 15 J was released during the impact?
A) 15 J; B)20 J; C) 30 J; D) 45 J
4. How will the momentum of a body change when its kinetic energy doubles?
A) will increase 2 times; B) will decrease by half;
B) will decrease by times;G) will increase in times.
5. Two plasticine balls fly towards each other. The moduli of their impulses are, respectively, 5 ∙ 10 - 2 kg ∙ m / s and 3 ∙ 10 - 2 kg ∙ m / s. After an inelastic impact, the impulse is:
A) 8 ∙ 10 - 2 kg ∙ m / s; B) 4 ∙ 10 - 2 kg ∙ m / s;
B) 2 ∙ 10 - 2 kg ∙ m / s; D) ∙ 10 - 2 kg ∙ m / s.
6. The figure shows a setup assembled to measure the speed of a bullet. If a bullet of mass m hits a block of mass M and gets stuck in it, then the block rises to a height h. How to determine the speed of a bullet v 0?
A) by the formula;
B) solving the system of equations
C) this setting does not allow finding v 0, since the law of conservation of momentum is not fulfilled when the bullet and the bar interact;
D) this setting does not allow finding v 0, since when the bullet and the bar interact, the law of conservation of mechanical energy is not fulfilled.
Answer: 3
Answer: 2
9. The kinetic energy of the body is 8 J, and the magnitude of the impulse is 4 N ∙ s. Body weight is equal to:
A) 0.5 kg; B) 1 kg; B) 2 kg; D) 32 kg
Solution to the problem of part C
Detailed solution
1. How to use the law of conservation of momentum?
Consider the states of the balls immediately before and immediately after impact. Since at the moment of impact the sum of external forces (gravity and thread tension) acting on the system is zero, the momentum of the system remains constant (the law of conservation of momentum)
In projection onto the Ox axis: p = - p 1 + p 2
2. How to use the law of conservation of energy?
According to the condition, the impact is absolutely elastic, therefore, the law of conservation of mechanical energy is fulfilled. And, since the potential energy before the impact is equal to the potential energy after the impact, the kinetic energy of the system has not changed either.
E kin = E kin1 + E kin2
3. How to compose and solve a system of equations?
Let us express the kinetic energy in terms of momentum:
Then, according to the law of conservation of energy
Multiply this expression by 2m:
We will square the equation p = - p 1 + p 2: p 2 = p 1 2 - 2 p 1 p 2 + p 2 2 and substitute it into the previous equality:
p 1 2 - 2 p 1 p 2 + p 2 2 =
From here
Answer:
Homework
Problem 1
Short solution to the problem:
Problem 2
Problem 3
Problem C2, 2009
Two balls, the masses of which differ by 3 times, hang in contact on vertical threads. The light ball is deflected at an angle of 90 ° and released without initial velocity. Find the ratio of the momentum of a light ball to the momentum of a heavy ball immediately after an absolutely elastic central collision.
1. The figure shows a load suspended on a thread and performing free vibrations like a pendulum. Within what limits does its potential energy change during these fluctuations of the load? The total mechanical energy of the load at the moment of deviation from the equilibrium position is 10 J.
A) Potential energy does not change and is equal to 10 J;
B) Potential energy does not change and is equal to 5 J;
C) Potential energy varies from 0 to 10 J;
D) Potential energy varies from 0 to 5 J.
3. The ball hit the wall, and the speed of the ball immediately after the hit is half its speed immediately before the hit. What is the kinetic energy of the ball before the impact, if the amount of heat released during the impact is 15 J?
A) 15 J; B) 20 J; C) 30 J; D) 45 J
Question: Why, when solving the problem, do we use only the conservation of the kinetic energies of the body?
4. How will the momentum of a body change when its kinetic energy doubles?
A) will increase by 2 times; B) will decrease by half;
C) will decrease by times; D) will increase in times.
5. Two plasticine balls fly towards each other. The moduli of their impulses are, respectively, 5 ∙ 10 - 2 kg ∙ m / s and 3 ∙ 10 - 2 kg ∙ m / s. After an inelastic impact, the impulse is:
9. The kinetic energy of the body is 8 J, and the magnitude of the impulse is 4 N ∙ s. Body weight is equal to:
A) 0.5 kg; B) 1 kg; B) 2 kg; D) 32 kg
Problem 1
Problem 2
Problem 3
The part of mechanics in which movement is studied without considering the reasons that cause this or that character of movement is called kinematics.Mechanical movement called a change in the position of the body relative to other bodies
Frame of reference the reference body, the associated coordinate system and the clock are called.
Reference body the body is called, relative to which the position of other bodies is considered.
Material point is called a body, the dimensions of which can be neglected in this problem.
Trajectory is called a mental line, which, in its movement, is described by a material point.
According to the shape of the trajectory, the movement is divided into:
a) straightforward- the trajectory is a straight line segment;
b) curvilinear- the trajectory is a segment of the curve.
Way 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 the path is different from moving. 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 point of destination (it does not matter at all which route this movement was made by). And the path is a scalar value that reflects the length of the trajectory traversed.
Uniform rectilinear movement is called a movement in which a material point for any equal intervals of time makes the same movements
The speed of uniform rectilinear movement is called the ratio of the movement to the time during which this movement occurred:
For uneven movement, use the concept average speed. Average speed is often entered as a scalar. This is the speed of such a uniform movement in which the body travels the same path in the same time as in the uneven movement:
Instant speed called the speed of the body at a given point of the trajectory or at a given moment in time.
Equally accelerated rectilinear motion- this is a rectilinear motion, in which the instantaneous speed for any equal time intervals changes by the same amount
The dependence of the coordinates of the body 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 is called uniformly accelerated motion with constant acceleration g = 9.8 m / s 2 independent of the mass of the falling body. It only happens under the influence of gravity.
Free fall speed is calculated using the formula:
Vertical displacement is calculated by the formula:
One of the types of motion of a material point is circular motion. With such a movement, the body's speed is directed along a tangent drawn to the circle at the point where the body is (linear velocity). You can describe the position of a body on a circle 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 the rotation of the circle radius 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 in a 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 during which the body describes a complete revolution is called period of circulation. The reciprocal of the period - the frequency of circulation - ν
Since with uniform motion around a circle the speed module does not change, but the direction of speed changes, there is an acceleration during such a motion. He's 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 reasons that caused the acceleration of bodies is called dynamics
Newton's first law:
There are such frames of reference relative to which the body keeps its speed constant or at rest, if other bodies do not 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 acting on it is called inertia. The phenomenon of conservation of the speed of a body with balanced external forces is called inertia. Inertial reference frames are called systems in which the first Newton's law is fulfilled.
Galileo's principle of relativity:
in all inertial reference frames under the same initial conditions, all mechanical phenomena proceed in the same way, i.e. obey the same laws
Weight is a measure of body inertia
Force is a quantitative measure of the interaction of bodies.
Newton's second law:
The force acting on the body is equal to the product of the body's mass and the acceleration imparted by this force:
$ F↖ (→) = m⋅a↖ (→) $
The addition of forces consists in finding 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 one straight line, equal in magnitude and opposite in direction:
$ F_1↖ (→) = -F_2↖ (→) $
III Newton's 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 Fig.).
Or, in short, the force of action is equal to the force of reaction. The question often arises: why does a horse pull the sled if these bodies interact with equal forces? This is possible only due to interaction with the third body - the Earth. The force with which the hooves hit the ground must be greater than the friction force of the sled against 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 force and deformation are directed in different directions.
When bodies move relative to each other, forces arise that impede movement. These forces are called frictional forces. Distinguish between static friction and sliding friction. Sliding friction force calculated by 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.
Rest friction occurs if the bodies do not move relative to each other. The static friction force can vary from zero to a certain maximum value
Gravitational forces are called the forces with which any two bodies are attracted to each other.
The law of universal gravitation:any two bodies are attracted to each other with a force 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 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 the 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 magnitude to the force of gravity:
If the body moves vertically with acceleration, then its weight will change.
When the body moves with acceleration upward, its weight
It is 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 body weight is less than the weight of the resting body.
Weightlessness is called such a motion of a body in which its acceleration is equal to the acceleration of gravity, 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 - the force of gravity, directed towards the center of the Earth.
First space speed- this is the speed that must be imparted 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 cosmic 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: steady, unstable and indifferent. If, when removing a body from a position of balance, forces arise that tend to return this body back, this stable balance. If forces arise that tend to take the body even further 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 equality to zero of the sum of the forces acting on the body, it is necessary that the algebraic sum of the moments of all forces acting on the body is equal to zero.
Here d is the shoulder of force. Shoulder of strength d is the distance from the axis of rotation to the line of action of the force.
Lever equilibrium condition:
the algebraic sum of the moments of all forces rotating the body is equal to zero.
By pressure is called 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 it is true Pascal's law:
pressure spreads in all directions unchanged.
If a liquid or gas is in the field of gravity, then each layer located above presses on the lower ones, and as it sinks into the liquid or gas, the pressure increases. For liquids
where ρ is the density of the liquid, h is the depth of penetration into the liquid.
A 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 set 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. The pistons have different areas. If some 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 transform the magnitude of the force. Since the pressure under the pistons must be the same, then 
Then A1 = A2.
On a body immersed in a liquid or gas, from the side of this liquid or gas, an upward buoyant force acts, which is called by the power of Archimedes
The magnitude of the buoyant force is set Archimedes' law: a body immersed in a liquid or gas is subjected to a buoyant force 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 burial - the volume of the submerged part of the body.
Body Swimming 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 impulse is called a physical quantity equal to the product of body weight by its speed:Momentum is a vector quantity. [p] = kg · m / s. Along with the impulse of the body is often used impulse of power. This is the product of force for the duration of its action.
The change in the momentum of a body is equal to the impulse of the force acting on this body. For an isolated system of bodies (a system whose bodies interact only with each other), momentum conservation law: 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 is called a physical quantity, which 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 displacement:
Power is the work done per unit of time:
The body's ability to do work is characterized by a quantity that is called energy. Mechanical energy is divided into kinetic and potential. If the body can do work due to its movement, it is said that it has kinetic energy. The kinetic energy of the translational motion of a material point is calculated by the formula
If the body can do work by changing its position relative to other bodies or by changing the position of body parts, it has potential energy. An example of potential energy: a body raised above the ground, its energy is calculated using the formula
where h is the lifting height
Compressed spring energy:
where k is the coefficient of spring stiffness, 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, it is true mechanical energy conservation law: if friction forces do not act between the bodies of an isolated system (or other forces leading to energy dissipation), 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 converted into internal energy.
1.5. Mechanical vibrations and waves
Fluctuations are called movements that have varying degrees of repetition in time. Oscillations are called periodic if the values of physical quantities that change during the oscillation 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 quantity A, equal to the largest absolute value of the oscillating physical quantity x, is called vibration amplitude... The expression α = ωt + ϕ determines the value of x at a given time and is called the oscillation phase. Period T is called the time during which the oscillating body makes one complete oscillation. The frequency of periodic fluctuations call the number of complete vibrations per unit of time:
Frequency is measured in s -1. This unit is called hertz (Hz).
A mathematical pendulum is called a material point of mass m, suspended on a weightless inextensible thread and vibrating in a vertical plane.
If one end of the spring is fixed motionlessly, and some body of mass m is attached to its other end, then when the body is removed from the equilibrium position, the spring will stretch and oscillate the body 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 vibrations in elastic media.
The wave is called transverse if the particles of the medium vibrate in directions perpendicular to the direction of wave propagation. The wave is called longitudinal if the vibrations of the particles of the medium occur in the direction of wave propagation.
Wave length is the distance between the two nearest points oscillating in the same phase:
where v is the speed of wave propagation.
Sound waves called waves, which oscillate 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 waves are called, the frequency of oscillations in which exceeds 20,000 Hz. Ultrasonic waves are not perceived by the human ear.
