$2.8.1$ The figure shows structures that hold a load weighing 10 kg. The cables are represented by thin lines, the rod is represented by a double line. Determine the tension force of the cables for case a and the force acting on the rod from the side of the cable thrown over it for case b.
Let's consider the following figure
$a$. Applying Newton Second Law for $x$-direction $$T_2~\cos{30^\circ}-T_1~\cos{30^\circ}=0$$ $$T_1=T_2 \;(1)$$ for $y$-direction $$(T_1+T_2)~\sin{30^\circ} = mg \;(2)$$ Let be $T_1=T_2=T$, so, back to $(2)$ $$2T~\sin{30^\circ}=mg$$ $$T = \frac{mg}{2~\sin{30^\circ}} = mg = \boxed{98~\rm{N}}$$ $b$. According figure, $$F^2 = T^2+T^2 = 2T^2$$ We consider the same tension $T$ because we suppose the threads are unextensible. $$F = \sqrt{2}~T \;(3)$$ Applying Newton Second Law $$T = mg \;(4)$$ Putting $(4)$ into $(3)$ $$F = \sqrt{2}~mg = \boxed{138~\rm{N}}$$ Note: Calculations were made considering $g$ = 9.8 N/kg and $\sqrt{2}$ = 1.41.
BSc. Luis Daniel Fernández Quintana
Physics Department (FCNE)
Universidad de Oriente, Cuba