A washer of mass m. On the way of a trolley of mass m sliding on a smooth horizontal table with a speed
pendulum thread length l\u003d 1 m, to which a load of mass m= 0.1 kg,
deflected by an angle a from the vertical position and released.
Thread tension force T at the moment the pendulum passes the equilibrium position is 2 N.
What is equal to the angle a?
Solution
Based on Newton's second law, acceleration
caused by the sum of the forces of gravity acting on the load and the tension of the thread,
when passing through the equilibrium position is equal to the centripetal acceleration:
According to the law of conservation of mechanical energy, for the weight of the pendulum
(the lower position of the load is chosen as the origin of the potential energy): 
Answer in general view and in numerical form:
Task 2
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 washer 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 = 4 m.
Find the mass of the puck m.
Ignore air resistance.
Solution
Task 3 (for independent solution)
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.
Task. A puck of mass m slides at a speed v 0 along a smooth horizontal surface of the table, hits a wedge of mass 2m at rest, slides along it without friction and separation, and leaves the wedge (see Fig.). The wedge, which did not come off the table, acquires the speed v 0 /4. Find the angle of inclination to the horizon of the surface of the upper part of the wedge. The lower part of the wedge has a smooth transition to the table surface. Ignore the change in the potential energy of the puck in the gravitational field as it moves along the wedge. Solution. 1. The work of non-conservative forces in the system is zero, therefore, the mechanical energy of the system is conserved (see Reference Note III, paragraph 10 vkotov.narod.ru/3.pdf) The kinetic energy of the washer before contact with the wedge (we take the potential energy of the washer in this position equal to zero) The kinetic energy of the puck immediately after leaving the wedge. The kinetic energy of the wedge immediately after the puck breaks off. (Change in the potential energy of the puck is neglected by the condition)
2. All external forces acting on the bodies of our system (gravity and the reaction force of the table) are perpendicular to the horizontal axis OX, therefore, the projection of the system's momentum on this axis is preserved (see Reference abstract III, p. 5 vkotov.narod.ru/3. pdf) Projection of the momentum of the washer before contact with the wedge. Projection of the momentum of the puck immediately after the release from the wedge. Projection of the momentum of the wedge immediately after the puck breaks off. 3. The speed modulus of the puck immediately after leaving the wedge v is related to the projections of this speed vx and vy: v 2 = vx 2 + vy 2 = (v 0 2 /4) + vy 2 Substitute this into the formula for the law of conservation of energy (point 1) and after contractions we get: After contraction we get: vx = v 0 /2 4. In the moving reference frame X"O"Y" associated with the wedge, the speed of the puck immediately after the separation v" will be directed at an angle to the horizon. The speed v" of the puck relative to the wedge is related to the speed v of the puck relative to the table according to the law of addition of speeds (see Reference Note I, p. 2 vkotov.narod.ru/1.pdf) The speed of the wedge immediately after the puck comes off v k \u003d v 0 / 4 .
О X Y 5. Let's perform the addition of vectors v "and v k according to the triangle rule and in the same figure we will show the decomposition of the vector v into components v x and v y: The required angle can be found from a triangle, the hypotenuse of which is v", and the legs are parallel to the axes OX and OY. The figure shows that tg v y /(v x v k) Substituting v x from point 2, v y from point 3 and v k from the problem data, we get the answer:
Option No. 2819169
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H(see picture).
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Ball, mass m 1 moving at speed v 1 , hits another ball, mass m 2. The impact is inelastic. Immediately after the impact, the speed of the balls is v. Find the amount of energy Δ U released upon impact.
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A puck is thrown down a smooth inclined plane. The maximum removal of the washer from the line of intersection of the inclined plane and the horizontal is 68 cm. The angle of the plane with the horizontal is α = 30°. Angle between initial speed and line ABβ = 60°. Find the initial speed of the puck.
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Two small bodies are thrown vertically upwards from the same point after a time interval Δ t= 3 s, informing them of the same initial speeds V 1 = V 2 = 20 m/s. At what height H bodies collide? Air resistance can be neglected.
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A rod slides along a horizontal plane AB, and the point O- its middle - has at a given time a speed equal in absolute value to 3 m/s and directed along the rod from the point A to the point B. Dot V while the rod has a speed modulo 5 m/s. What is the speed of the point and how is it directed A at this point in time?
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r 0 \u003d 1 cm starts to beat upwards from the hose with a speed v 0 = 20 m/s. Find the jet radius r on high h= 16 m vertically from the end of the hose. Ignore friction and surface tension forces, consider the velocity of water particles along the vertical in any cross section of the jet to be the same for a given section, and the particles themselves to be in a state of free fall in the gravity field.
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Projectile with mass 2 m breaks in flight 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 the value Δ E. The modulus of the speed of a fragment moving in the direction of the projectile is equal to and the modulus of the speed of the second fragment is Find Δ E.
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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 impact are mutually opposite and equal, and the mass of the bar is 4 times the mass of the plasticine. The coefficient of sliding friction between the bar and the table How far will the sticky bars with plasticine move by the moment when their speed decreases by 2 times?
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The horizontal surface is divided into two parts: smooth and rough. On the border of these parts is a small cube. From the side of the smooth part, a ball of mass M= 200 g moving at speed v 0 = 3 m/s. Determine the mass of the cube m, if it stops after an absolutely elastic central impact with a ball at a distance L= 1 m from the collision site. The coefficient of friction of the cube on the surface μ = 0.3.
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A small ball with a mass is suspended on a light inextensible thread with a length that breaks under tension. The ball is removed from the equilibrium position (it is shown in the figure by a dotted line) and released. When the ball passes the equilibrium position, the thread breaks, and the ball immediately collides absolutely inelastically with a block of mass lying motionless on a smooth horizontal surface of the table. What is the speed u bar after impact? Assume that the block moves forward after the impact.
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To the ceiling on two identical light springs with a total stiffness k= 400 N/m suspended cup mass m= 500 g. From a height h= 10 cm a load of the same mass falls into the cup and sticks to it m(see fig.). What is the maximum distance H after that, will the cup descend relative to its original position? Neglect mechanical energy losses.
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In the "Flying Cyclist" trick, the rider moves down the ramp under the influence of gravity, starting from rest at a height H(see picture).
At the edge of the springboard, the speed of the rider is directed at an angle 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 trampoline? Ignore air resistance and friction.
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m= 1 kg with speed v= 5 m/s. He experiences a windshield absolutely elastic collision with another ball of mass M= 2 kg, which was at rest before the collision (see Fig.). After that, the second ball hits a massive piece of plasticine glued to the plane and sticks to it. How much heat was released in the process of sticking the second ball to a piece of plasticine?
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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 an amount Determine the speed of the fragment flying in the direction of the projectile.
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On the smooth horizontal surface of the table rests a hill with two peaks, the heights of which are and (see & nbsp picture). On the right top of the hill is the puck. From a slight push, the puck and the slide begin to move, 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. The speed of the puck on the left top of the hill turned out to be equal to v. Find the ratio of the masses of the puck and slide.
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A block of mass slides down an inclined plane from a height h and, moving along a horizontal surface, collides with a stationary bar mass. As a result of an absolutely inelastic collision, the total kinetic energy of the bars becomes equal to 2.5 J. Determine the height of the inclined plane h. Ignore friction during motion. Assume that the inclined plane smoothly turns into a horizontal one.
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On the smooth horizontal surface of the table rests a hill with two peaks, the heights of which h and 4 h(See figure). On the right top of the hill is the puck. The mass of the slide is 8 times the mass of the puck. 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 puck on the left top of the hill.
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In the "Flying Cyclist" trick, the rider moves on a smooth ramp under the influence of gravity, starting from rest at a certain height (see figure). At the edge of the springboard, the rider's speed is directed at an angle α = 60° to the horizon. Having flown through the air, it lands on a horizontal table, rising in flight to a height h over the edge of the trampoline. From what height H did the racer start?
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The high-speed electric train "Nevsky Express" traveled from St. Petersburg to Moscow along a straight horizontal track at a speed of v= 180 km/h. The passenger of the train hung a plumb line in front of him and began to follow his behavior. At some point, the train began to brake with constant acceleration in order to stop at Bologoy. At the beginning of deceleration, the plumb line deviated by a certain maximum angle α, and then it oscillated with a slowly decreasing amplitude until the train stopped. What was the angle α if the distance to the stopping point at the time of the start of braking was 2.5 km?
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Thin homogeneous rod AB hinged at a point A Sun m= 1 kg, its angle of inclination to the horizon α = 30°. Find the modulus of force acting on the rod from the side of the hinge. Make a drawing on which indicate all the forces acting on the rod.
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H(see picture). At the edge of the springboard, the rider's speed is directed at an angle α = 60° to the horizon. After flying through the air, it lands on a horizontal table at the same height as the edge of the springboard. What is the maximum possible flight height for a rider?
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In the "Flying Cyclist" trick, the rider moves on a smooth ramp under the influence of gravity, starting from rest at a height H(see picture). At the edge of the springboard, the rider's speed is directed at an angle α = 60° to the horizon. After flying through the air, it lands on a horizontal table at the same height as the edge of the springboard. What is the flight distance of the rider?
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On the smooth horizontal surface of the table rests a hill with two peaks, the heights of which are and (see & nbsp picture). On the right top of the hill is the puck. 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. The speed of the puck on the left top of the hill turned out to be equal. Find the ratio of the masses of the puck and the hill.
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At the interface between two immiscible liquids having densities ρ 1 \u003d 900 kg / m 3 and ρ 2 = 3ρ 1, the ball floats (see picture). What should be the density of the ball ρ so that one third of its volume is above the liquid interface?
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On a smooth horizontal plane there are two identical ideally elastic smooth washers. One of them moves with a speed modulo 3 m/s, and the other is at rest near a straight line drawn through the center of the first puck in the direction of its speed. The washers collide, and after the collision the second, initially at rest, bounces off at an angle α = 30° to this line. Find the speed of the first puck after the collision.
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A hill with two peaks, the heights of which h and 3 h, rests on a smooth horizontal surface of the table (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.
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A slide with two peaks, the heights of which rest on the smooth horizontal surface of the table (see & nbsp picture). There is a coin on the right top of the slide. From a slight push, the coin and the hill begin to move, and the coin moves to the left, without breaking away from the smooth surface of the hill, and the progressively moving hill does not come off the table. At some point in time, the coin was on the left top of the hill, having a speed Find the speed of the hill at this moment.
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The horizontal surface is divided into two parts: smooth and rough. At the boundary of these parts there is a cube of mass m= 100 g. From the side of the smooth part, a metal ball of mass M= 300 g moving at speed v 0 = 2 m/s. Determine the distance L, which the cube will pass until it stops after an absolutely elastic central collision with the ball. The coefficient of friction of the cube on the surface μ = 0.3.
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The pendulum consists of a small mass M\u003d 100 g hanging on a light inextensible thread of length L= 50 cm. It hangs at rest in a vertical position. A small body of mass hits the load and sticks to it m= 20 g flying in the horizontal direction. As a result, the pendulum rotates in a vertical plane around its suspension point, and the pendulum's weight moves all the time in a circle, making a full turn. What could be the speed of the body before the impact?
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Two balls whose masses m= 0.1 kg and M\u003d 0.2 kg, hang, touching, on vertical threads of the same length l(see picture). The left ball is deflected through an angle of 90° and released with an initial velocity equal to zero. As a result of an absolutely inelastic impact of balls, the amount of heat is released Q\u003d 1 J. Determine the length of the threads l.
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The Sapsan high-speed electric train traveled along a straight horizontal track at a speed v= 180 km/h. The passenger of the train hung a plumb line in front of him and began to follow his behavior. At some point, the train began to slow down with constant acceleration in order to stop in Tver. At the same time, the plumb line at the beginning of braking deviated to the maximum angle α = 5.7°, and then it fluctuated with a slowly decreasing amplitude until the train stopped. At what distance L from the station in Tver "Sapsan" began braking?
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A small ball falls from above onto an inclined plane and is elastically reflected from it. The angle of inclination of the plane to the horizon is How far horizontally does the ball move between the first and second hits on the plane? The speed of the ball immediately before the first impact is directed vertically downwards and is equal to 1 m/s.
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Thin homogeneous rod AB hinged at a point A and held by a horizontal thread Sun(see picture). The friction in the hinge is negligible. Rod weight m= 1 kg, its angle of inclination to the horizon α = 45°. Find the modulus of force acting on the rod from the side of the hinge. Make a drawing on which indicate all the forces acting on the rod.
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A weight is attached to a vertical spring with a stiffness of 400 N/m. The system is in equilibrium. At a certain point in time, part of the load is unhooked and the system again comes into equilibrium, while the spring is displaced by 3 cm. Determine the mass of the unhooked part of the load.
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In the "Flying Cyclist" trick, the rider moves on a smooth ramp under the influence of gravity, starting from rest at a certain height (see figure). At the edge of the springboard, the rider's speed is directed at an angle α = 60° to the horizon. Flying through the air, he landed on a horizontal table at the same height as the edge of the springboard. The flight range of the rider is S. At what height H Is there a starting point above the edge of the springboard?
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Projectile with mass 2 m, moving at a speed 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 the value Δ E. The speed of a fragment moving in the direction of the projectile is Find Δ E.
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A ball of mass m= 2 kg with speed v= 2 m/s. It experiences a head-on absolutely elastic collision with another ball of mass M= 2.5 kg, which was at rest before the collision (see Fig.). After that, the second ball hits a massive piece of plasticine glued to the plane and sticks to it. Find the modulus of momentum that the second ball transferred to a piece of plasticine.
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