As shown in the figure, a beam is supported by two pillars separated by a distance $ell$. The beam has a mass $M$, a length $L$, and a girl of mass $m$ is walking from the left end toward the right.
(a) Find a symbolic expression for the normal force exerted on the beam by the pillar on the right, when the beam is on the verge of tipping. (Use the following as necessary: $M$, $m$, and $g$.)
$N_2 = $
(b) Find a symbolic expression for the girl's position, when the beam is on the verge of tipping. (Use the following as necessary: $M$, $m$, $L$ and $ell$.)
$x = $
(c) Find a symbolic expression for the minimum value of $ell$ that will allow the girl to reach the end of the beam without it tipping. (Use the following as necessary: $M$, $m$, and $L$.)
$ell_{min} = $