$2.1.25.$ A thread spanned over a block with a fixed axis is passed through a slot. At the ends of the thread are suspended loads, the mass of which is $m_1$ and $m_2$. Determine the acceleration of loads if a constant frictional force $F_{tr}$ acts on the thread from the side of the slot when it moves.
1. Due to the weightlessness and non-stretchability of the thread, as well as the ideal properties of the block (no losses and low weight), the problem can be solved in the following approximation) $$ a_1=a_2=a $$ $$ T_1=T_2=T $$ 2. The equations of motion of loads in projection onto the vertical axis in this case are written as follows: $$ \left\{\begin{matrix} m_1a = m_2g - T \\ m_2a =T-m_2g-F_{fr} \end{matrix}\right. $$ 3. Solving the equations together, we obtain $$ a=\frac{(m_1-m_2)g-F_{fr}}{m_1+m_2} $$ 4. Substituting the acceleration value into the first equation of the system allows us to determine the tension of the thread $$ \boxed{T=m_1\frac{2m_2g+F_{fr}}{m_1+m_2}} $$
$$T=m_1\frac{2m_2g+F_{fr}}{m_1+m_2}$$