Solution D1-07 (Figure D1.0 condition 7 S.M. Targ 1989)

Solution to problem D1-07 (Figure D1.0 condition 7 S.M. Targ 1989)

Given a load of mass m, which received an initial speed v0 at point A and moves in a curved pipe ABC located in a vertical plane. Pipe sections can be inclined or horizontal (see Fig. D1.0 - D1.9, Table D1). In section AB, the load is acted upon by a constant force Q (its direction is shown 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 can be neglected.

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 acted upon 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 given in the table.

In the calculations, we assume that the load is a material point and the distance AB = l or the time t1 of movement of the load from point A to point B is known. It is necessary to find the law of movement of the load on the section BC, i.e. x = f(t), where x = BD.

Answer:

In section AB, the load is acted upon by a constant force Q and a resistance force of the medium R, which depends on the speed v of the load and is directed against the movement. Using Newton's second law, we can write the equation of motion of the load in section AB:

m*a = Q - R,

where a is the acceleration of the load.

Since the friction of the load on the pipe in section AB is negligible, the friction force is zero. The drag force of the medium can be expressed as follows:

R = k*v,

where k is the resistance coefficient of the medium.

Thus, the equation for the movement of cargo in section AB will take the form:

ma = Q - kv.

Solving this equation, we obtain the law of load movement in section AB:

v = (Q/k) + C1exp(-kt/m),

where C1 is the integration constant, which can be found from the initial conditions of the problem. Since at point A the load has an initial speed v0, then C1 = (v0 - Q/k). Substituting C1 into the equation, we get:

v = (v0exp(-kt/m)) + (Q/k)(1 - exp(-kt/m)).

In the section BC, the load is acted upon by a friction force and a variable force F, the projection of which Fx on the x axis is given in the table. Using Newton's second law, we can write the equation of motion of the load on the aircraft section:

ma = Fx - fN,

where N is the normal force acting on the load from the pipe.

Since the load is moving along an inclined surface, the normal force can be expressed as follows:

N = mgcos(a),

where g is the acceleration of gravity, α is the angle of inclination of the surface.

Thus, the equation for the movement of cargo on the aircraft section will take the form:

ma = Fx - fmgcos(a).

Solving this equation, we obtain the law of cargo movement on the aircraft section:

x = (1/(2f))[(Fx/m) - gcos(a)]t^2 + (v0 + (Q/k))(1 - exp(-kt/m)) - (Q/k),

where t is the time of cargo movement on the aircraft section.

Thus, we have obtained the law of cargo movement on the aircraft section, expressed in terms of coordinate x and time t. It depends on the initial conditions of the problem, such as the mass of the load, the initial speed, the coefficient of friction and the forces acting on the load. By solving this problem, it is possible to determine the nature of the movement of the load in a given section of the pipe.

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This solution describes the movement of a load of mass m in a curved pipe, which is subject to various forces, such as gravity, friction and environmental resistance. The solution to the problem is presented in the form of mathematical equations and expressions that will help you better understand the physical laws underlying the problem.

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The digital product “Solution D1-07 (Figure D1.0 condition 7 S.M. Targ 1989)” is a solution to problem D1-07 from the textbook “Collection of problems in general physics” by author S.M. Targa, published in 1989. This problem considers the movement of a load of mass m in a curved pipe, which is subject to various forces, such as gravity, friction, and the drag force of the medium.

The solution to the problem is presented in the form of mathematical equations and expressions that will help you better understand the physical laws underlying the problem. In particular, the solution uses Newton’s second law to write the equations of motion of the load in sections AB and BC, and the friction coefficient of the load on the pipe f = 0.2 is taken into account when describing the movement in section BC.

By purchasing the digital product "Solution D1-07 (Figure D1.0 condition 7 S.M. Targ 1989)", you receive a convenient and easily accessible tool for studying physics. The solution to the problem is presented in the form of an HTML page with colorful illustrations and formulas in LaTeX. Our store guarantees product quality and fast delivery by email.

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Solution D1-07 is a problem from the textbook by S.M. Targa "Motion of a material point along a curved pipe." In this problem, a load of mass m moves along a curved pipe ABC, which is in a vertical plane. In section AB, the load is acted upon by a constant force Q and the resistance force of the medium R, as well as the force of gravity. At point B, the load passes to section BC of the pipe, where, in addition to the force of gravity, it is acted upon by the friction force and the variable force F.

The task is to find the law of cargo movement on the aircraft section, i.e. function x=f(t), where x is the distance between point B and the load, and t is the time of movement of the load on the aircraft section. To solve the problem, it is necessary to know the mass of the load, the initial speed v0, the coefficient of friction of the load on the pipe f, the projection of the variable force Fx on the x axis and the distance AB=l or the time t1 of movement of the load from point A to point B.

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