Savchenko Solutions

Solutions of Savchenko Problems in Physics

Aliaksandr Melnichenka
October 2023

    <h3 id="back-link"><a href="/#2.8">$\leftarrow$Back</a></h3>

    <h3> Statement </h3>

    <p>
      $2.8.4$
      Through three holes made in a desktop pass threads united by a common knot. At the free ending of each thread is hanged equal loads. Determine the angles between threads. There is no friction.  
    </p>
    <center>
      <figure>
        <img src="statement.png"
          loading="lazy" alt="2.8.4" width="200" />
        <figcaption>
          For problem 2.8.4
        </figcaption>
      </figure>
    </center>

    <h3>Solution</h3>
    <p>
      Tension in each thread is the same ($T=mg$). Moreover, note that the absolute value of the sum of two any tensions must be equal to the third one which is also $T$ and be directed in opposed sense for maintain the equilibrium. So, consider for example the angle $\gamma$
    </p>
    
    <center>
      <figure>
        <img src="draw.png"
          loading="lazy" alt="2.8.4" width="300" />
        <figcaption>
          Top view
        </figcaption>
      </figure>
    </center>

    <p>
      $$T_R = T = 2T\cos{\frac{\gamma}{2}}$$
      Hence
      $$\cos{\frac{\gamma}{2}} = 0.5$$
      $$\gamma = \frac{2\pi}{3} = 120^\circ$$
      But, since the above condition of the sum of two any tensions always is true, the three angles must be equal.
      $$\boxed{\alpha=\beta=\gamma=120^\circ}$$
    </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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