Solution to problem 2.2.7 from the collection of Kepe O.E.

2.2.7 In triangle ABC, a right angle at vertex C, forces F1, F2 and F3 act on vertices A, B and C, respectively. It is necessary to determine the value of angle C in degrees at which the main moment M0 = -2 kN•m for a given system of forces, if it is known that F2 = 4 kN and distance l = 1 m. The answer is angle C equal to 30.0 degrees.

To solve this problem it is necessary to use the moment of force. The moment of force is defined as the product of the force modulus and the distance between the line of action of the force and the reference point. In this case, to determine the angle C, it is necessary to draw up an equation for the equilibrium of moments of forces relative to point C:

F1 * a + F2 * l * sin(С) - F3 * b = 0,

where a and b are the distances between points C and B, C and A, respectively.

From the problem conditions it is known that F2 = 4 kN and l = 1 m. Let us find the values ​​of a and b:

a = l * cos(С) = 1 * cos(С), b = l * sin(С) = 1 * sin(С).

Let's substitute the values ​​of a and b into the equilibrium equation of moments of forces and solve it with respect to angle C:

F1 * l * cos(С) + F2 * l * sin(С) * sin(С) - F3 * l * sin(С) = 0, F1 * cos(С) + 4 * sin^2(С) - F3 * sin(С) = 0, F1 * cos(С) + 4 - F3 * sin(С) / sin(С) = 0, F1 * cos(С) + 4 - F3 = 0, F1 * cos(С ) = F3 - 4.

Since M0 = -2 kN•m, then

М0 = F1 * a + F2 * l * sin(С) - F3 * b = F1 * l * cos(С) + F2 * l * sin(С) * sin(С) - F3 * l * sin(С) = -2.

From the equation of equilibrium of moments of forces we find F1:

F1 = (F3 - 4) / cos(С).

Let's substitute the value of F1 into the equation M0 and solve it with respect to angle C:

F1 * l * cos(С) + F2 * l * sin(С) * sin(С) - F3 * l * sin(С) = -2, (F3 - 4) * l * cos(С) / cos( C) + 4 * l * sin^2(C) - F3 * l * sin(C) = -2, (4 - F3) * l * sin(C) = 2, sin(C) = 2 / (4 - F3) = 0.5, C = arcsin(0.5) = 30.0 degrees.

Thus, the angle C at which the main moment of this system of forces M0 = -2 kN•m is equal to 30.0 degrees.

Solution to problem 2.2.7 from the collection of Kepe O.?.

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Solution to problem 2.2.7 from the collection of Kepe O.?. is to determine the angle? in degrees, at which the main moment of the force system M0 = -2 kN•m.

To solve a problem it is necessary to use moments of forces. The moment of force is defined as the product of force and the perpendicular distance from the point of application of the force to the axis of rotation. In this problem, the axis of rotation is the point of intersection of the medians of the triangle.

According to the conditions of the problem, three forces are applied to the vertices of a right triangle, the force F2 = 4 kN and the distance l = 1 m are known. It is necessary to find the angle? such that the main moment of the force system M0 is equal to -2 kN•m.

To solve the problem, it is necessary to find the moments of each of the forces relative to the point of intersection of the medians of the triangle and add them. Then it is necessary to solve the equation relating the sum of moments to the angle ?.

The solution to this problem is described in detail in the collection of Kepe O.?. The answer to the problem is 30.0 degrees.

Problem 2.2.7 from the collection of Kepe O.?. belongs to the section "Probability Theory and Mathematical Statistics". To solve it, it is necessary to apply combinatorics and probability formulas.

The problem states that in a group of 20 people, 3 people are selected at random. It is necessary to find the probability that among those selected there will be at least one group leader, if it is known that there are 2 leaders in the group.

To solve the problem, it is necessary to use the combinatorics method and the conditional probability formula. It is necessary to determine the total number of combinations of 20 people of 3, and then the number of combinations in which at least one leader is present. After this, you can calculate the probability of the desired event.

Solution to problem 2.2.7 from the collection of Kepe O.?. will help you understand the use of combinatorial formulas and conditional probability to solve problems in probability theory and mathematical statistics.

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