scalar equations for net torque & force ...

$T_1 r - T_2 r - \tau_f = \dfrac{1}{2}m_p r^2 \cdot \dfrac{a}{r}$

$T_1 - T_2 - \dfrac{\tau_f}{r} = \dfrac{1}{2}m_p \cdot a$

$\color{red}{T_1 - T_2 - \dfrac{25}{3} = a}$

$\color{red}{4g - T_1 = 4a}$

$\color{red}{T_2 - 2g = 2a}$

summing the last

three equations yields ...

$2g - \dfrac{25}{3} = 7a \implies a = \dfrac{6g-25}{21} \, m/s^2$

$|\Delta y| = \dfrac{1}{2}at^2 \implies t = \sqrt{\dfrac{2\Delta y}{a}} = \sqrt{\dfrac{42}{6g-25}} \approx 1.1 \, sec$