A uniform rod AB having mass 2m and length $2\mathcal{l}$  is kept on a smooth horizontal plane. This rod is orthogonally connected to a horizontal taut string whose other end has a connected particle of mass m. If particle is given initial horizontal velocity $v_0$, perpendicular to string, then just after giving this velocity, the tension in the string is $\frac{m{v}_0^2}{n\mathcal{l}}$ . Then the value of $n$  is