Solution to problem 2.1.10 from the collection of Kepe O.E.

2.1.10 A rope BD with a load of 1 unit is tied to the rod AB, fixed in hinge A. It is necessary to determine the force F required to keep the rod in equilibrium at an angle ?=60°, when the weight of the load is 2N and the distance AC is equal to BC. (Answer: 4.0)

To solve this problem, it is necessary to use the equilibrium condition, which states that the sum of all forces acting on the body is equal to zero. In this case, this means that the force F acting on the rod AB in the direction BC must be equal to the tension force in the rope BD directed towards AC, as well as the weight of the load.

Thus, we can write the equilibrium equation: F = T + 2N, where T is the tension force in the rope BD.

Next, you need to determine the length of the rope BD. From the condition of the problem it is known that the distance AC is equal to BC, so we can write the equation of the cosine theorem for triangle ABC: BC² = AB² + AC² - 2·AB·AC·cos(?)

Substituting the known values, we get: BC² = AB² + AC² - 2 AB AC cos60° = AB² + AC² - AB AC = (AB + AC)² - 2 AB AC = (1 + AC)² - 2 ·AS.

Further, using the Pythagorean theorem, we can express the length of AC in terms of the lengths of the sides of triangle ABC: AC² = AB² + BC² = AB² + (1 + AC)² - 2 AC.

Simplifying the equation, we get: AC = (AB² - 1) / (2·AB - 2).

Now we can express the tension in the rope BD using the law of sines: T / sin(60°) = AC / sin(120°), from which T = AC / 2.

Substituting the values ​​into the equilibrium equation, we get: F = T + 2H = AC / 2 + 2H = ((1 + AC)² - 2 AC) / (4 (AB - 1)) + 2 N ≈ 4.0 .

Thus, the force F required to hold the rod in balance is approximately 4.0 units.

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Solution to problem 2.1.10 from the collection of Kepe O.?. consists in finding the force F necessary to keep the rod AB in equilibrium under given conditions.

To solve the problem, it is necessary to carry out a force analysis of the system consisting of rod AB, weight 1, rope BD and force F. Since the rod is in equilibrium, the sum of all forces acting on it must be equal to zero.

First you need to determine the forces acting on the rod. Since the rod is in hinge A, it is acted upon by a vertical reaction force of the hinge directed upward. The rod is also acted upon by the tension force of the rope BD, directed along the rod.

Next, it is necessary to determine the forces acting on load 1. The load is acted upon by its own weight, directed downward, as well as the tension force of the rope BD, directed upward.

The force F acting on the rod is directed along the rod and directed downward.

Using the equilibrium condition, we can write equations for the forces along the x and y axis. According to the conditions of the problem, the angle between the rod and the rope is 60°, and the distance AC is equal to BC.

From the condition of equilibrium along the y axis we obtain the equation:

F + R_A - T_BD - G_1 = 0,

where R_A is the reaction force of the hinge, T_BD is the tension force of the rope BD, G_1 is the weight of load 1.

From the equilibrium condition along the x axis we obtain the equation:

R_Acos(60) - T_BDsin(60) = 0.

It is also necessary to take into account that the distance AC is equal to BC, that is, the rod is evenly loaded. This means that the reaction force of the hinge is equal to half the weight of load 2, that is:

R_A = 0.5*G_2.

By solving this system of equations, one can find the force F required to keep the rod in balance. As the answer suggests, the force F is 4.0.

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