Problem $12\mathrm{.}068-$ Tension in a cord connecting a collar and a drum

The $4\mathrm{.}00-kg$ collar $B$ slides on the frictionless arm $A{A}^{\text{'}}.$ The arm is attached to drum $D$

and rotates about $O$ in a horizontal plane at the rate ${}^{},$ where ${}^{}$ and $t$ are expressed in

$ra\frac{d}{s}$ and seconds, respectively. As the arm$-$drum assembly rotates, a mechanism within

the drum releases cord so that the collar moves outward from $O$ with a constant speed

of $0\mathrm{.}5\frac{m}{s}.$ Knowing that at $t=0,r=0,$ determine the time at which the tension in

the cord is equal to the magnitude of the horizontal force exerted on $B$ by arm $A{A}^{\text{'}}.$

The time at which the tension in the cord is equal to the magnitude of the horizontal

force exerted by the arm is

S$.$