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A beam of rectangular cross section $($width $b$ and height $h)$ supports a uniformly distributed load along its entire length $L.$ The allowable stresses in bending and shear are $$ and $,$ respectively.

$($a$)$ If the beam is simply supported, what is the span length $L_1$ below which the shear stress governs the allowable load and above which the bending stress governs?

$($b$)$ If the beam is supported as a cantilever $($with one end fixed$),$ what is the length $L_2$ below which the shear stress governs the allowable load and above which the bending stress governs?

$($v$)$