Solution to problem 20.2.14 from the collection of Kepe O.E.

In this problem, there is cylinder 1, to which a pair of forces is applied with a moment M = 120 N•m and a moment of friction forces Mtr = 10 N•m. A load 2 with a mass of m2 = 40 kg is attached to the end of an inextensible thread. The radius of the cylinder is R = 0.3 m.

To solve the problem, we choose the angle φ as the generalized coordinate. Then the moment of inertia of the cylinder will be equal to I = mR²/2, where m is the mass of the cylinder. Taking this into account, the equation of motion of the load can be written as:

m2gRsinφ - T = m2R²φ''

where g is the gravitational acceleration, T is the generalized force, m2R²φ'' is the angular acceleration of the load.

Since the thread is inextensible, the speed of the load is equal to the speed of the point of contact of the thread with the cylinder, which means that the speed of the load can be defined as Rφ'. Also taking into account that the moment of inertia of the cylinder is equal to mR²/2, we obtain the following expression for the moment of friction forces:

Mtr = - (mR²/2)φ'

Taking this into account, we express the generalized force T:

T = m2gRsinφ + (mR²/2)φ'' - Мтр = m2gRsinφ + (mR²/2)φ'' + (mR²/2)φ'

Solving this equation, we obtain the generalized force T = -7.72.

Thus, we determined the generalized force based on the given parameters of the system.

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Problem 20.2.14 from the collection of Kepe O.?. is formulated as follows:

Cylinder 1 is acted upon by a pair of forces with a moment of $M=120$ N$\cdot$m and a moment of frictional forces $M_{\text{tr}}=10$ N$\cdot$m. A load 2 with a mass of $m_2=40$ kg is attached to the cylinder, tied to the end of an inextensible thread. The radius of the cylinder is $R=0.3$ m. By choosing the angle $\theta$ as the generalized coordinate, it is necessary to determine the generalized force.

The solution to this problem is associated with determining the equation of motion of the system. To do this, it is necessary to express the acceleration of the load and the cylinder through a generalized coordinate and then write dynamic equations for each of the elements of the system.

As a result of solving this problem, the value of the generalized force is obtained, which is equal to $-7.72$.

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