Two bodies of the same mass connected by a thread slide along an inclined plane. Thread tension force T. There is no friction between one body and the board. Determine the friction force between the board and another body.

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    <h3> Statement </h3>
    <p>
        $2.1.47^*.$ Two bodies of the same mass connected by a thread slide along an inclined plane. Thread tension force $T$. There is no friction between one body and the board. Determine the friction force between the board and another body.

    <h3>Solution</h3>
    <p>
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Body of mass $m_3$ moves with acceleration of the center of mass of the system

Considering the system as a whole (not taking into account internal forces) $$\boxed{a_3 = \frac{F}{M+2m}}$$ In this case, a force equivalent to double the tension force of the thread $2T$ acts on the body $m_3$. $$2T = M a_3$$

$$T = F\frac{M}{2(M+2m)}$$
The body $m_1$ is acted upon by two forces $\vec{T}$ and $\vec{F}$

We write Newton's second law for the first body as $$F-T = ma_1$$

$$\boxed{a_1=\frac{F(M+4m)}{2m(M+2m)}}$$
Similarly, the second body $m_2$ is acted upon only by $\vec{T}$
$$m_2 = T$$

$$\boxed{a_2=\frac{FM}{2m(M+2m)}}$$

    <h4>Answer</h4>
    <p>
        Acceleration of cargoes 1–3:

$$a_1=\frac{F(M+4m)}{2m(M+2m)}$$

$$a_2=\frac{FM}{2m(M+2m)}$$

$$a_3=\frac{F}{M+2m}$$

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