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.

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    <h3> Statement </h3>
    <p>
        $2.1.25.$ 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.

    <h3>Solution</h3>
    <p>
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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
$$\boxed{T=m_1\frac{2m_2g+F_{fr}}{m_1+m_2}}$$

    <h4>Answer</h4>
    <p>
        $$T=m_1\frac{2m_2g+F_{fr}}{m_1+m_2}$$
    </p>


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