Dievsky V.A. - Solving problem D4 option 14 task 2

To solve the problem of equilibrium of the mechanical system presented in the figure, we will use the Lagrange principle. 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 . Unnumbered blocks and rollers are considered weightless, and friction on the axes of the drum and blocks can be neglected.

First, let's determine the acceleration of the load a. The figure shows that the load is in a state of equilibrium, which means that the sum of all forces acting on it is equal to zero:

ΣF = 0

where ΣF is the total force.

Let us depict on the diagram all the forces acting on the load:

F is the required tension force in the cable; G - cargo weight; T1 and T2 - tension in cables thrown over blocks; N1, N2, N3 and N4 - support reaction forces.

Let's create the equations of motion for the load along the x axis:

ΣFx = max = 0

where m is the mass of the load, akh is the acceleration of the load along the x axis.

Summing up all the forces acting on the load, we get:

F - T1 - T2 - fN3 = max

Let's create the equations of motion for the load along the y-axis:

ΣFy = may = 0

where ay is the acceleration of the load along the y-axis.

Summing up all the forces acting on the load, we get:

N1 + N2 + G - N4 - fN3 = 0

Let's create the equations of motion for block 1:

ΣF1 = ma1 = 0

where a1 is the acceleration of block 1.

Summing up all the forces acting on block 1, we get:

T1 - N1 - fN3 = ma1

Let's create the equations of motion for block 2:

ΣF2 = ma2 = 0

where a2 is the acceleration of block 2.

Summing up all the forces acting on block 2, we get:

T2 - N2 - fN4 = ma2

Let's create the equations of motion for the drum:

ΣF3 = ma3 = 0

where a3 is the acceleration of the drum.

Summing up all the forces acting on the drum, we get:

F - 2T1 - 2T2 - M/R2 - fN2(r2/R2) = ma3

Thus, we have obtained a system of equations that must be solved for the desired force F. The value of F at which the mechanical system will be in equilibrium can be determined from the equation ΣFx = 0. In this case, the maximum value of the force F will correspond to the case when the friction force reaches its limit value.

Dievsky V.A. - Solution to problem D4 option 14 task 2 - this is a digital product that is presented in a digital goods store. This product contains a solution to a physics problem using Lagrange's principle. Solving the problem allows us to determine the magnitude of the force F at which the mechanical system will be in equilibrium. The product contains the initial data, as well as a system of equations that must be solved to determine the required force F.

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Solution of problem D4 option 14 task 2 Dievsky V.A. is a useful digital product for students and anyone interested in physics. It will help you better understand Lagrange's principle and apply it in practice when solving problems in physics.

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This product is a problem from the textbook "Problems in General Physics. Volume 1. Mechanics" edited by V.A. Dievsky. Solution of problem D4-14, option 14, task 2.

In the task, it is necessary to determine the magnitude of the force F at which the mechanical system presented in the diagram will be in equilibrium, taking into account friction. To solve the problem it is necessary to use the Lagrange principle.

Input data for the problem: 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. Unnumbered blocks and rollers are considered weightless, and friction on the axes of the drum and blocks can be neglected.

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