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 horizontal, 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 along it. This happened at a board tilt angle of 14 0. Why coefficient is equal friction?
46. The load moves up an inclined plane (angle of inclination a to the horizontal) with constant acceleration and under the influence of a force parallel to the inclined plane and coinciding in direction with the acceleration vector. By what amount D m should the coefficient of friction of the load on the plane be increased so that the body rises evenly?
47. A 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 along the 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 speed will the body have at the end of the journey?
48. An ice slide makes an angle a= 30 0 with the horizon. A stone is passed along it from bottom to top, which during t 1 = 2 s covers 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 equal 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 friction coefficient of the upper bar m is 2 times greater than the friction coefficient of the lower one.
50. A block slides from an inclined plane of length l and height h and then along a horizontal plane to a distance S, after which it stops. Determine the coefficient of friction of the block, considering it constant.
51. After what time will the speed of the body, to which the speed V 0 was imparted, directed upward along the inclined plane, be equal to V 0 again? Friction coefficient m , angle of inclination of the plane to the horizon a. The body begins to move with speed V 0, being in the middle of the inclined plane.
52. 
Two bars with a mass of 0.2 each were 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 is m 1 = 0.3, the upper one is m 2 = 0.1. Determine the force of interaction between the bars when they slide together from an inclined plane.
53. 
A flat plate of mass m 2 is placed on an inclined plane with an angle of inclination a, and a block m 1 is placed on it. The coefficient of friction between the block and the plate is m 1. Determine at what values the friction coefficient 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.
54. An inclined plane with inclination angle a moves with acceleration a. Starting from what value of acceleration 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 a car can move up along it with an acceleration of 0.6 m/s 2?
56. A block of mass 0.5 kg lies on a rough surface inclined to the horizontal at an angle a. What is the minimum horizontal force F, directed perpendicular to the plane of the drawing, that must be applied to the block in order for it to move? Friction coefficient m= 0.7.
57. 
On an inclined plane there is a mass of mass m 1 = 5 kg, connected by a thread thrown over a block to another mass m 2. The coefficient of friction between the first load 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?
58. Weightless block fixed on the top of two inclined planes making angles of 30 0 and 45 0 with the horizon. Kettlebells equal mass 1 kg each are connected with 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. Neglect friction in the block.
59. A puck thrown along an inclined plane slides along it, moving upward, then returns to the place where it was thrown. The dependence of the puck speed modulus on time is shown in Fig. Find the angle of inclination of the plane to the horizon and maximum height lifting the washer.
60. On an inclined plane with an inclination angle of 30 0, blocks m 1 = 1 kg, m 2 = 2 kg move as one whole (with the same acceleration). The friction coefficients 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 during movement.
61. *A body of mass m 1 rises along an inclined plane with acceleration a under the action of a force F parallel to the inclined plane and directed in the direction of motion of the body. By what amount 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.
62. A load of mass m moves freely down an inclined plane (angle of inclination a to the horizontal) with some constant acceleration. What force F parallel to the inclined plane and directed upward must be applied to the load so that it rises with the same acceleration? The friction coefficient is constant.
63. A load of mass m rises uniformly along an inclined plane under the action of some force parallel to the inclined plane and coinciding in direction with the direction of movement. By what amount D F should this force be increased so that the body rises with acceleration? The friction coefficient does not change.
64. On a smooth horizontal table lies a prism of mass M with an angle of inclination a, and on it a prism of mass m. The smaller prism is subject to a horizontal force F, while both prisms move along the table as one unit (i.e. without changing relative position). Determine the friction force between the prisms.
65. From a point lying at the upper end of the vertical diameter of a certain vertical circle, a point body begins to slide along a groove installed along a chord making an angle a with the vertical. How long will it take for the load to reach the lower end of the chord? Circle diameter D.
Start typing part of the condition (for example, can , what is equal to or find ):
MECHANICS. CHAPTER II. BASICS OF DYNAMICS. Movement on an inclined plane
- No. 2821. A load weighing 26 kg lies on an inclined plane 13 m long and 5 m high. The friction coefficient is 0.5. What force must be applied to the load along the plane in order to pull the load? to steal a load?
- No. 283. What force must be applied to lift a trolley weighing 600 kg along a trestle with an inclination angle of 20°, if the coefficient of resistance to movement is 0.05?
- No. 284. During 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 upward 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. Using a dynamometer located parallel to the plane, the block was first pulled up the inclined plane and then pulled down. Find the difference between the dynamism readings
- No. 286*. To hold the cart on an inclined plane with an angle of inclination α, it is necessary to apply a force F1 directed upward along the inclined plane, and to lift it upward, 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 horizontal. At what values of the friction coefficient 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 mass of 50 kg. What force directed along the plane must be applied to hold this load? pull up evenly? pull with an acceleration of 1 m/s2? Friction coefficient 0.2.
- No. 289. A car weighing 4 tons is moving 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 weighing 3000 tons is moving down a slope of 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 began to move 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 s after it started moving, if it passed a horizontal section of the road in 3
- No. 292(n). A block of mass 2 kg is placed on an inclined plane with an angle of inclination of 30°. What force, directed horizontally (Fig. 39), must be applied to the block so that it moves uniformly along the inclined plane? Coefficient of friction between a block and an inclined plane
The movement of a body along an inclined plane is classic example body movements under the influence of several undirected forces. Standard method solving problems of this kind of motion consists in decomposing the vectors of all forces into components directed along coordinate axes. Such components are linearly independent. This allows us to write Newton's second law for components along each axis separately. Thus, Newton's second law, which is vector equation, turns into a system of two (three for the three-dimensional case) algebraic equations.
The forces acting on the block are
case of accelerated downward movement
Consider a body that is sliding down an inclined plane. In this case, the following forces act on it:
- Gravity m g , directed vertically downwards;
- Ground reaction force N , directed perpendicular to the plane;
- Sliding friction force F tr, directed opposite to the speed (up along the inclined plane when the body slides)
When solving problems in which an inclined plane appears, it is often convenient to introduce an inclined coordinate system, the OX axis of which is directed downward along the plane. This is convenient, because in this case you will have to decompose only one vector into components - the gravity vector m g
, and the friction force vector F
tr and ground reaction forces N
already directed along the axes. With this expansion, the x-component of gravity is equal to mg sin( α
) and corresponds to the “pulling force” responsible for accelerated movement down, and the y-component is mg cos( α
) = N balances the ground reaction force, since there is no body movement along the OY axis.
Sliding friction force F tr = µN proportional to the ground reaction force. This allows us to obtain the following expression for the friction force: F tr = µmg cos( α
). This force is opposite to the "pulling" component of gravity. Therefore for body sliding down
, we obtain expressions for the total resultant force and acceleration:
F x = mg(sin( α
) – µ
cos( α
));
a x = g(sin( α
) – µ
cos( α
)).
It's not hard to see what if µ < tg(α ), then the expression has a positive sign and we are dealing with uniformly accelerated motion down an inclined plane. If µ >tg( α ), then the acceleration will have negative sign and the movement will be equally slow. Such movement is possible only if the body is given starting speed in a downslope direction. In this case, the body will gradually stop. If provided µ >tg( α ) the object is initially at rest, it will not begin to slide down. Here the static friction force will completely compensate for the “pulling” component of gravity.
When the friction coefficient is exactly equal to tangent plane inclination angle: µ = tg( α ), we are dealing with mutual compensation of all three forces. In this case, according to Newton's first law, the body can either be at rest or move with constant speed(Wherein uniform motion only downwards possible).
The forces acting on the block are
sliding on an inclined plane:
case of slow motion upward
However, the body can also drive up an inclined plane. An example of such motion is the movement of a hockey puck up an ice slide. When a body moves upward, both the frictional force and the “pulling” component of gravity are directed downward along the inclined plane. In this case, we are always dealing with uniformly slow motion, since total force directed in the opposite direction to the speed. The expression for acceleration for this situation is obtained in a similar way and differs only in sign. So for body sliding up an inclined plane , we have.