The $6\mathrm{.}00-kg$ 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 rad$/$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 $A{A}^{\text{'}}.$