I am confused with determining the net force on the climbing man below by the rope. My analysis is as follows. Let the tension be $T$. As both hands hold the rope then there are $T+T=2T$ upward acting on the man. Of course there is a $mg$ downward.
My question: What is the correct net tension force acting on the man?
The tension is not the same for all parts of the rope. If the tension is $T_1$ between the hands and $T_2$ between the top hand and the ceiling then.
$$ T_2 = T_1 + F_2 \\ T_1 = F_1 $$
where $F_1+F_2=W$ are the forces acted upon the arms. You arrive at this if you make two free body diagrams, one at each hand.
The result is that $T_2 = W$ and $T_1=F_1<W$. You have to establish the load distribution between the hands $$\gamma = \frac{ F_1}{F_1+F_2} = \frac{F_1}{W}$$ to show that the rope tension between the hands is $T_1 = \gamma W$, $F_1 = \gamma W$ and $F_2 = (1-\gamma) W$.
