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

    <h3> Statement </h3>

    <p>
      $4.2.11$
      With what force a heavy stick presses on the bottom of the reservoir, if a hollow ball of radius $r$, rigidly connected to the stick, is half immersed in the liquid?  The density of the liquid is $\rho$, the length of the stick is $l$.      
    </p>
    <center>
      <figure>
        <img src="statement.png"
          loading="lazy" alt="4.2.11" width="200" />
        <figcaption>
          For problem 4.2.11
        </figcaption>
      </figure>
    </center>

    <h3>Solution</h3>
    <p>
      Let's consider the following figure
    </p>
    <center>
        <figure>
          <img src="draw.png"
            loading="lazy" alt="4.2.11" width="200" />
          <figcaption>
            Force analysis
          </figcaption>
        </figure>
      </center>
    <p>
      The system is on equilibrium in above position, so total torque is zero. Considering as a pivot point the center of mass of the stick (located at stick's half),
      $$F_A\left(\frac{l}{2}+r\right)\sin{\left(\frac{\pi}{2}-\alpha\right)}+mg\left(\frac{l}{2}+r\right)\sin{\left(\frac{\pi}{2}+\alpha\right)}+N\frac{l}{2}\sin{\left(\frac{\pi}{2}+\alpha\right)}=0$$
      $$F_A\left(\frac{l}{2}+r\right)\cos{\alpha}-mg\left(\frac{l}{2}+r\right)\cos{\alpha}-N\frac{l}{2}\cos{\alpha}=0$$
      $$F_A\left(\frac{l}{2}+r\right)-mg\left(\frac{l}{2}+r\right)-N\frac{l}{2}=0$$
      $$\rho\~g\~\frac{V}{2}\left(\frac{l}{2}+r\right)-mg\left(\frac{l}{2}+r\right)-N\frac{l}{2}=0$$
      Separating $N$, and considering that ball's mass can be expressed as $m=\rho_0 V$ and ball's volume is $V=\frac{4}{3}\pi r^3$
      $$N = \frac{2\pi r^3g}{3}\left(1+2\frac{r}{l}\right)(\rho-2\rho_0)$$
      But, since the ball is a hollow sphere, it can be considered massless, so $\rho_0 = 0$, then
      $$\boxed{N = \frac{2\pi r^3\rho g}{3}\left(1+2\frac{r}{l}\right)}$$
    </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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