Determine the pressure force of the logs of mass m on the walls of the channel. The upper log is half submerged in water, and the lower log touches the upper part of the water surface. The logs are equal.

Solutions of Savchenko Problems in Physics

Aliaksandr Melnichenka
October 2023

    <h3 id="back-link"><a href="/#4.2">$\leftarrow$Back</a></h3>

    <h3> Statement </h3>

    <p>
      $4.2.14$
      Determine the pressure force of the logs of mass $m$ on the walls of the channel.  The upper log is half submerged in water, and the lower log touches the upper part of the water surface. The logs are equal.
    </p>
    <center>
        <figure>
          <img src="statement.png"
            loading="lazy" alt="4.2.14" width="300" />
          <figcaption>
            For problem 4.2.14 
          </figcaption>
        </figure>
      </center>

    <h3>Solution</h3>
    <p>
      Let's consider the following figure
    </p>
    <center>
        <figure>
          <img src="draw.png"
            loading="lazy" alt="4.2.14" width="550" />
          <figcaption>
            Force analysis
          </figcaption>
        </figure>
      </center>
    <p>
      From right figure
      $$\cos{\theta} = \frac{R}{2R} = 0.5$$
      so, $\theta = 60^\circ$. Applying Newton Second Law for submerged trunk on $x$-axis,
      $$N = N_t\~\sin{\theta} \\;(1)$$
      on $y$-axis
      $$F_{A1} = mg + N_t\~\cos{\theta}$$
      $$\rho_w\~g\~V = mg + N_t\~\cos{\theta} \\;(2)$$
      For half-submerged trunk, on $x$-axis,
      $$N = N_t\~\sin{\theta}$$
      on $y$-axis
      $$F_{A2} + N_t\~\cos{\theta} = mg$$
      $$\rho_w\~g\~\frac{V}{2} + N_t\~\cos{\theta} = mg \\;(3)$$
      Substituting $(2)$ into $(3)$, it is obtained
      $$N_t = \frac{mg}{3\~\cos{\theta}} \\;(4)$$
      Putting $(4)$ into $(1)$
      $$N = \frac{mg}{3}\~\tan{\theta}$$
      $$\boxed{N = \frac{mg}{\sqrt{3}}}$$
      Note: You can prove that both pressure forces generated by trunks are equals.
    </p>
    <p style="text-align: right; font-style: italic; font-size: 14;">   
      BSc. Luis Daniel Fernández Quintana<br>
      Physics Department (FCNE)<br>
      Universidad de Oriente, Cuba<br>
    </p>



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