44. How long will it take for a body of mass m to slide off inclined plane height h, inclined at an angle a to the horizon, if it moves uniformly along an inclined plane with an angle of inclination b?.

45. To determine the coefficient of friction between wooden surfaces, a block was placed on a board and one end of the board was raised until the block began to slide on it. This happened at an angle of inclination of the board 14 0 . What is equal to the coefficient friction?

Here 1 is the acceleration of the body. Using the formula of the second problem, we get that the speed of the body at the end of the path is equal to. Because the final result does not depend on the angle of inclination, the same formula is also valid for a body moving along the second inclined plane through an angle of 1.

Using the Pythagorean theorem, we get the result we are looking for. The fourth problem has already been discussed in §. It is necessary to decompose the initial velocity of the body into its horizontal and vertical components. Or, finally, the third problem. The law of conservation of energy has the form.

This is a common position that not all students understand. Now let's consider a specific case about the work of a centripetal force. The body is moving at a constant speed, so its kinetic energy does not change: the first channel is closed. The body moves in a horizontal plane, therefore, it also does not change its potential energy: the second channel is closed. The organism in question does not work on any other body: it has closed the third channel. Now you must determine your position.

46. ​​The load moves up the inclined plane (the angle of inclination a to the horizon) with a constant acceleration a under the action of a force parallel to the inclined plane and coinciding in direction with the acceleration vector. To what extent D m should the coefficient of friction of the load on the plane be increased so that the body rises evenly?

47. The body lies on an inclined plane making an angle of 45 0 with the horizon. a) At what limiting coefficient of friction will the body begin to slide down an inclined plane? b) With what acceleration will the body slide along the plane if the coefficient of friction is 0.03? c) How long will it take to travel 100 m under these conditions? d) What is the speed of the body at the end of the journey?

Consider two communicating vessels connected by a tube with a narrow key. We open the key, and the liquid flows out of the container from left to right. We calculate the potential energy of the liquid in the initial and final states by multiplying the weight of the liquid in each vessel by half the height of the liquid column.

If in the initial state potential energy equal, then. in the final state it is the same. Thus, in the final state, the potential energy of the liquid is two times lower than in the initial state. He asks himself: what did the other half of the energy do?

48. An ice hill makes an angle a = 30 0 with the horizon. A stone is thrown along it from bottom to top, which during t 1 \u003d 2 s travels a distance of 16 m, after which it rolls down? What is the coefficient of friction between the slide and the stone?

49. Two bars with the same masses are fastened with a thread and are on an inclined plane with an angle of inclination a. Determine the tension of the thread T when the bars move along an inclined plane, if the coefficient of friction of the upper bar m is 2 times greater than the coefficient of friction of the lower one.

If after that we use the law of conservation of energy and place in the result, we get. The first of these equations expresses the law of conservation of momentum and the second law of conservation of energy. This shows that the greater the mass M, the more energy into heat. For example, the collision of two steel spheres with a good degree of approximation can be considered as a perfectly elastic collision. In this case, there is a fully elastic deformation of the spheres, and there is no heat release. There is no reason to worry about the "perversion" of these laws, since in absolutely elastic bodies the collisions move at different speeds.

50. A bar slides from an inclined plane of length l and height h and further along the horizontal plane to a distance S, after which it stops. Determine the coefficient of friction of the bar, assuming it to be constant.

51. After what time will the speed of the body, to which the speed V 0 was reported, directed upward along the inclined plane, again be equal to V 0? Friction coefficient m , the angle of inclination of the plane to the horizon a. The body begins to move at a speed V 0 , being in the middle of an inclined plane.

If, after a perfectly inelastic collision, the colliding bodies move at the same speed, then after an elastic collision, each body moves at its own speed, and two equations are required to find two unknowns. Let us assume that as a result of the collision, the first body deviates backward.

In this case, the laws of conservation of momentum and energy can be written as follows. However, in general, you don't have to worry about the direction of travel. We analyzed special case Center Impact: The spheres move before and after the collision along a line through their centers. A more general case of a "non-central" shock will be considered below. just for different types collisions, there are various forms of equations to describe the conservation laws.

52. Two bars of mass 0.2 each are placed on an inclined plane with an angle of 45 0 as shown in the figure. The coefficient of friction of the lower bar on the inclined plane m 1 = 0.3, the upper m 2 = 0.1. Determine the force of interaction of the bars during their joint sliding from an inclined plane.

The class of shocks we studied is an extreme case. In real cases, a certain amount of heat is always released, and as a result, colliding bodies can separate at different speeds. However, a number of real cases can be described quite well using these simple models: absolutely elastic and absolutely inelastic. Let's look at an example of an off-center elastic impact. A body lies on a horizontal plane, having an inclined plane with an inclination angle of 45 °.

It should not be forgotten that the law of conservation of momentum is a vector equality, since the amount of movement is a vector quantity that has the same direction as the speed. It is true that in the case when all velocities are directed along the same line, this vector equality can be written in scalar form. This is exactly what we did when we reached the central shock. In the general case, it is necessary to decompose the velocities into two mutually perpendicular axes and write the momentum conservation law separately for each of the axes.

53. A flat plate of mass m 2 is placed on an inclined plane with an angle of inclination a, and a bar m 1 is placed on it. The coefficient of friction between the bar and the plate m 1 . Determine at what values ​​the coefficient of friction m 2 between the slab and the plane, the slab will not move if it is known that the block is sliding on the slab.

In this problem, we choose a horizontal and vertical axis. In the horizontal direction, the momentum conservation law has the following aspect. It is easy to understand that another body is acting in this problem: the Earth. That is, if the Earth did not act, the body M would not move horizontally, after! shock.

Thus, the presence of the Earth in our problem does not change an aspect of the equation, but leads to an equation that expresses the law of conservation of the amount of motion in the vertical direction. If we multiply this value by zero, we get zero again. From this we conclude that the Earth takes part in this problem in a very peculiar way: when it receives a certain amount of motion, it practically does not receive energy. In other words, the Earth interferes with the law of conservation of momentum and does not interfere with the law of conservation of energy.

54. An inclined plane with an angle of inclination a moves with an acceleration a. Starting from what value of acceleration a will a body lying on an inclined plane begin to rise? The coefficient of friction between the body and the inclined plane is m.

55. What should be the minimum coefficient of friction between the tires and the surface of an inclined road with a slope of 30 0 so that the car can move up it with an acceleration of 0.6 m / s 2?

A body of 3 kg mass falls from a certain height with an initial velocity of 2 ms directed vertically downwards. The resistance force is assumed to be constant. The body slides: first along an inclined plane at an angle of 30°, and then continues to move along the horizontal plane. Determine the coefficient of friction if it is known that the body moves in the horizontal plane at the same distance as in the inclined plane. Calculate the useful working coefficient of an inclined plane when a body slides uniformly along it.

Air resistance is neglected. Find the driving speed together with the vehicle and the platform just after the automatic latch is triggered. Calculate the distance traveled by the vehicle and platform after engagement if the drag force is 5% of the weight. At what height do the spheres rise after the collision, yes. shock is perfectly elastic, shock is absolutely inelastic? What should be the minimum velocity of the projectile so that after hitting the sphere it reaches a full cycle in the vertical plane?

56. A block of mass 0.5 kg lies on a rough surface inclined to the horizon at an angle a. What is the minimum horizontal force F, directed perpendicular to the plane of the drawing, to act on the bar to move it. Friction coefficient m= 0.7.

57. On an inclined plane there is a load of mass m 1 = 5 kg, connected by a thread thrown over a block with another load m 2 . The coefficient of friction between the first weight and the plane is 0.1. The angle of inclination of the plane to the horizon is 37 0 . At what values ​​of mass m 2 will the system be in equilibrium?

Previous knowledge gained by the teacher with the student

Strategies and class features

At this stage, it should be clear how the slope of the plane can increase the speed of the body located in the plane to slide. The teacher can start the discussion by asking the class about the following situation. A boy in a wheelbarrow or a skateboard and wants to go down a sloping street, if he is still new to the sport, which street would he be most afraid of? Answer: On a sloping street, the speed at the end of the ramp will be higher, and this can lead to a fall. This causes the boy to steeply increase his speed in the same amount of time. Answer: As we now choose, the component of the force that contributes to the increase in speed is proportional to the angle between the plane and the horizontal.

Now the teacher has to present the slide simulator from the following site

To start the simulation, the teacher must complete the following steps. Watch well and repeat the process by clicking on restore then chart and then unlock and look closely at the speed chart. Observe well and repeat the process by clicking restore, then draw the chart, and then unlock and look closely at the speed chart. Compare the graphs obtained in points 1 and ask the students in a group to discuss in which of the situations the speed increased faster? Give an explanation for this difference depending on the slope. After the group discussion, students should share the results achieved by the group with other students in the class. Each group should have its own time and after submitting reports, the teacher should complete the class that caused the most speed increase. In the end, it is checked that the greater the slope of the plane, i.e. the greater the angle between the plane and the horizontal, the greater the increase in speed over a period of time. Thus, a larger angle implies more acceleration.

Analysis of forces acting on a body moving in an inclined plane

  • Answer: Of course, there will be more fear on the steepest street.
  • Why is a beginner afraid on the steepest street?
  • Set the angle to 10, the coefficient of friction to 0, and the mass.
  • Then unlock by clicking the lock and seeing the movement.
  • Increase the angle to 20, keep the coefficient of friction 0 and mass.
To begin studying the forces acting on a body located on a tilt plane, display the following animation.

58. A weightless block is fixed on top of two inclined planes making angles of 30 0 and 45 0 with the horizon. Weights of equal mass of 1 kg are connected by a thread and thrown over a block. Find: 1) the acceleration with which the weights move; 2) thread tension. The friction coefficients of weights on inclined planes are 0.1. Ignore friction in the block.

59. A puck thrown along an inclined plane slides along it, moving up, then returns to the place of the throw. The plot of the puck speed modulus versus time is shown in fig. Find the angle of inclination of the plane to the horizontal and the maximum height of the puck.

Repeat this animation several times and ask the students in the group to explain the explanation for the following questions.

Ventura. At this stage, it is necessary to include the friction force and study the uniform motion and the inevitability of motion. The teacher should present another feature that demonstrates movement at a constant speed and corresponding forces. Environment educational object that simulates the situation of an inclined plane, the address image.

60. On an inclined plane with an angle of inclination of 30 0, bars m 1 \u003d 1 kg, m 2 \u003d 2 kg move as one (with the same acceleration). The coefficients of friction between the inclined plane and these bars are respectively equal to m 1 =0.25 and m 2 =0.10. Find the force R of interaction between the bars in the process of movement.

61. * A body of mass m 1 rises along an inclined plane with an acceleration a under the action of a force F parallel to the inclined plane and directed in the direction of the body's movement. To what extent D m should the body weight be increased so that it rises evenly? The coefficient of friction, the magnitude and direction of the force F does not change.

Using the simulator available on the lab computers or in the projection room, run some simulations that cause an analysis of the forces acting on the block as it moves. With the class divided into groups, have them complete the following steps and note the findings about what they have observed. The force required and the force of the parallel component that corresponds to the weight component. Watch the forces and pay attention to what you observe. The force required and the force of the parallel component, compare the figure with what appeared again. What is the black color component in the picture. Because it is against the movement. Compare the differences between situations 1 and 2 without friction and friction. Analyze the force diagram and explain what will change if only the angle is changed. Repeat the simulation, only changing the angle from 30 to 45 and then. After running simulations and group annotations, students should choose one representative from each group to communicate with their other results. This result can be presented to the whole class, with each group in turn. At the end, the teacher should make final considerations and indicate.

  • Increase the block level with the following parameters.
  • Make a block with new parameters.
When there is no friction force, the force required to drag the block at a constant speed along the ramp must be equal to the weight component, parallel to the ramp, parallel component.

62. A load of mass m moves freely down an inclined plane (the angle of inclination a to the horizon) with some constant acceleration. What force F parallel to the inclined plane and directed upwards must be applied to the load so that it rises with the same acceleration? The coefficient of friction is constant.

63. A load of mass m rises uniformly along an inclined plane under the action of a certain force parallel to the inclined plane and coinciding in direction with the direction of movement. To what extent D F should increase this force so that the body rises with acceleration a? The coefficient of friction does not change.

64. On a smooth horizontal table lies a prism of mass M with an angle of inclination a, and on it is a prism of mass m. A horizontal force F acts on the smaller prism, while both prisms move along the table as a whole (i.e. without changing relative position). Determine the force of friction between the prisms.

65. From a point lying on the upper end of the vertical diameter of a certain vertically located circle, a point body begins to slide along a chute installed along a chord making an angle a with the vertical. How long will it take for the weight to reach the lower end of the chord? Circle diameter D.

Start typing part of the condition (for example, can , equal to or find ):

MECHANICS. CHAPTER II. FOUNDATIONS OF DYNAMICS. Movement on an inclined plane

  • No. 2821. On an inclined plane 13 m long and 5 m high lies a load of 26 kg. The coefficient of friction is 0.5. What force must be applied to the load along the plane to pull the load? to carry a load?
  • No. 283. What force must be applied to lift a trolley weighing 600 kg along a flyover with an inclination angle of 20 °, if the coefficient of resistance to movement is 0.05?
  • No. 284. During the laboratory work, the following data were obtained: the length of the inclined plane is 1 m, the height is 20 cm, the mass of the wooden block is 200 g, the traction force when the block moves up is 1 N. Find the coefficient of friction.
  • No. 285. A block of mass 2 kg rests on an inclined plane 50 cm long and 10 cm high. With the help of a dynamometer located parallel to the plane, the bar was first pulled up the inclined plane, and then dragged down. Find the difference in the readings of the dynam
  • No. 286*. To keep the trolley on an inclined plane with an angle of inclination α, it is necessary to apply a force F1 directed upwards along the inclined plane, and to lift it up, it is necessary to apply a force F2. Find the drag coefficient.
  • No. 287. The inclined plane is located at an angle α = 30° to the horizon. At what values ​​of the coefficient of friction μ is it more difficult to pull a load along it than to lift it vertically?
  • No. 288. On an inclined plane 5 m long and 3 m high there is a load of 50 kg. What force, directed along the plane, must be applied to hold this load? pull evenly up? pull with an acceleration of 1 m/s2? Friction coefficient 0.2.
  • No. 289. A car with a mass of 4 tons moves uphill with an acceleration of 0.2 m/s2. Find the traction force if the slope1 is 0.02 and the drag coefficient is 0.04.
  • No. 290. A train with a mass of 3000 tons moves down a slope equal to 0.003. The coefficient of resistance to movement is 0.008. With what acceleration does the train move if the traction force of the locomotive is: a) 300 kN; b) 150 kN; c) 90 kN?
  • No. 291. A motorcycle weighing 300 kg started moving from rest on a horizontal section of the road. Then the road went downhill, equal to 0.02. What speed did the motorcycle acquire 10 seconds after the start of movement, if it passed a horizontal section of the road with
  • No. 292(n). A block of mass 2 kg is placed on an inclined plane with an inclination angle of 30°. What horizontal force (Fig. 39) must be applied to the bar so that it moves uniformly along the inclined plane? The coefficient of friction of the bar on the inclined plane