Two balls, A and B $,$ with masses $\backslash ($ m$\_\{\backslash$mathrm$\{$a$\}\}\backslash )$ and $\backslash ($ m$\_\{\backslash$mathrm$\{$b$\}\}\backslash )$ are connected by a taut, massless string, and are moving along a horizontal frictionless plane. The distance between the centers of the two balls is $\backslash ($ L $\backslash ).$ At a certain instant, the velocity of ball B has magnitude $\backslash ($ v $\backslash )$ and is directed perpendicular to the string and parallel to the horizontal plane, and the velocity of ball A is zero. $($Figure $1)$

Figure

Part A

Find $\backslash ($ T $\backslash ),$ the tension in the string.

Express $\backslash ($ T $\backslash )$ in terms of $\backslash ($ m$\_\{\backslash$mathrm$\{$a$\}\},$ m$\_\{\backslash$mathrm$\{$b$\}\},$ v $\backslash ),$ and $\backslash ($ L $\backslash ).$