$15\mathrm{.}2\mathrm{.}15$ A massless rigid rod with length $\backslash (\backslash$ell $\backslash )$ attached to a ball of mass $\backslash ($ M $\backslash )$ spins at a constant angular rate $\backslash (\backslash$omega $\backslash )$ which is maintained by a motor $($not shown$)$ at the hinge point. The rod can only withstand a tension of $\backslash ($ T$\_\{$c r$\}\backslash )$ before breaking. Find the maximum angular speed of the ball so that the rod does not break assuming

a$)$ there is no gravity, and

b$)$ there is gravity $($neglect bending stresses$).$

Problem $15\mathrm{.}2\mathrm{.}15$