Dievsky V.A. - Solving problem D4 option 8 task 2

D4-08 (Task 2) Dievsky For the diagram presented in the figure, it is necessary to determine the magnitude of the force F at which the mechanical system will be in equilibrium, using the Lagrange principle. In this case, the presence of friction should be taken into account and the maximum value of this force should be found. Initial data:

• load weight G = 20 kN,
• torque M = 1 kNm,
• drum radius R2 = 0.4 m (double drum also has r2 = 0.2 m),
• angle α = 300,
• sliding friction coefficient f = 0.5.

Unnumbered blocks and rollers are considered weightless. The friction on the axes of the drum and blocks can be neglected. To solve the problem, we will use the Lagrange principle, which allows us to find the equations of motion of a system based on its energy. In this case, the potential energy of the system will be equal to the work done by the force F moving the load, and the kinetic energy will be equal to the work done by the forces of gravity and friction. Using these data, it is possible to calculate the magnitude of the force F at which the system is in equilibrium. First, let's find the work of gravity and friction: where:

• G - load weight,
• R2 - drum radius,
• r2 - block radius (if there is one),
• α is the angle of inclination of the thread to the horizon,
• f is the sliding friction coefficient.

Substituting the values ​​and calculating, we get: Now let's find the potential energy of the system: To find the equation of motion of the system, it is necessary to calculate the Lagrangian: Let's calculate the derivative of the Lagrangian with respect to speed and solve the equation of motion: Answer: Thus, the maximum value of the force F at which the mechanical system will be in equilibrium , is 56.5 kN.

Dievsky V.A. - Solving problem D4 option 8 task 2

That digital product is a solution to problem D4, option 8, task 2, developed by Professor V.A. Dievsky. The solution to the problem is presented in the form of a detailed description of all the stages and steps necessary to solve it, which will allow you to easily understand the problem and get the desired result.

The solution to problem D4 option 8 task 2 is based on the application of the Lagrange principle to determine the magnitude of the force F at which the mechanical system will be in equilibrium. In solving the problem, such parameters as the weight of the load, torque, drum radius, angle of inclination of the thread to the horizon and sliding friction coefficient are taken into account.

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This product is a literary solution to problem D4 option 8 task 2, created by the author Dievsky V.A. The solution to the problem is based on the Lagrange principle and is aimed at determining the magnitude of the force F, which is necessary to achieve equilibrium of the mechanical system shown in the figure. The problem takes into account the presence of friction, and also indicates the initial data: load weight G = 20 kN, torque M = 1 kNm, drum radius R2 = 0.4 m (double drum also has r2 = 0.2 m), angle α = 300 and sliding friction coefficient f = 0.5. Blocks and unnumbered blocks are assumed to be weightless, and friction on the axes of the drum and blocks is neglected.

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