Solution D1-33 (Figure D1.3 condition 3 S.M. Targ 1989)

The solution to problem D1-33 (shown in Figure D1.3, condition 3 from the book by S.M. Targ, 1989) is as follows. A load of mass m, having received an initial speed v0 at point A, moves in a curved pipe ABC located in a vertical plane. Pipe sections can be inclined or horizontal (see Figures D1.0 - D1.9 and Table D1). In section AB, in addition to the force of gravity, the load is acted upon by a constant force Q (its direction is indicated in the figures) and a resistance force of the medium R, which depends on the speed v of the load and is directed against the movement. The friction of the load on the pipe in section AB is not taken into account. At point B, the load, without changing its speed, moves to the section BC of the pipe, where, in addition to the force of gravity, it is affected by the friction force (friction coefficient of the load on the pipe f = 0.2) and the variable force F, the projection of which Fx on the x axis is given in table. The load is considered as a material point. If the distance AB = l or the time t1 of movement of the load from point A to point B is known, then it is necessary to find the law of movement of the load on the section BC, i.e. x = f(t), where x = BD.

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Solution D1-33 is a problem about the movement of a load of mass m along a curved pipe ABC located in a vertical plane. The initial speed of the load at point A is v0. In section AB, the load is acted upon by the force of gravity, the constant force Q and the resistance force of the medium R, which depends on the speed of the load. In the section BC, the load moves at a constant speed, it is acted upon by the force of gravity, the force of friction and the variable force F, the projection of which Fx on the x axis is given in the table. The coefficient of friction between the load and the pipe is f = 0.2.

The task is to find the law of movement of the load on the aircraft section, i.e., a function that describes the dependence of the x coordinate of the load on time t. The distance between points A and B is equal to l, or the time of movement of the load from point A to point B is equal to t1.

To solve the problem, it is necessary to apply the equations of motion of a material point and the equation of motion with a variable force. After analytical transformations, it is possible to obtain the law of cargo movement on the aircraft section, i.e., the function x = f(t).

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