$4.2.14$ 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.
Let's consider the following figure
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.
BSc. Luis Daniel Fernández Quintana
Physics Department (FCNE)
Universidad de Oriente, Cuba