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.

Savchenko Solutions

Solutions of Savchenko Problems in Physics
knowledge must be free

    <h3 id="back-link"><a href="/#2.1">$\leftarrow$Back</a></h3>

    <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>
        <center>

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:

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

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

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

<footer class="row container">
  <br>
    <p>
        <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small>
    </p>
    <p>
        <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
    </p>
</footer>