$2.8.4$ 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.
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 $\gamma$
$$T_R = T = 2T\cos{\frac{\gamma}{2}}$$ 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. $$\boxed{\alpha=\beta=\gamma=120^\circ}$$
BSc. Luis Daniel Fernández Quintana
Physics Department (FCNE)
Universidad de Oriente, Cuba