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.
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<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.
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For problem 2.8.4
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<h3>Solution</h3>
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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 $\gamma$
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Top view
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<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.
$$\boxed{\alpha=\beta=\gamma=120^\circ}$$
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<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>
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