To solve this problem, we first need to figure out the inertia of the turbine and its angular acceleration (deceleration, actually) while being brought to rest. Then we can figure out the torque that needs to be applied for one minute to achieve that deceleration       .

If a turbine has a mass of 1200 kg, a radius of gyration of 0.8 m, then the inertia would be given by the following equation: Inertia = mk², where m is the mass of the turbine and k is the radius of gyration. In this case the inertia would equal 768 kg m².

We know that the final angular velocity = initial angular velocity + (angular acceleration)*t:

1. The initial angular velocity is 50 rev/min = 50 x 2π/60 = 5.236 radians/s

2. The final angular velocity is 0 rev/min = 0 radians/s

3. Therefore: 0 = 5.236 radians/s + (angular acceleration)*60 s

4. i.e. angular acceleration = -0.087267 radians/s² (note that this is negative, which is consistent with the fact that it the turbine would decelerate)

Torque is given by (Inertia)*(Angular acceleration) = 768 kg m²  * (-0.087267 radians/s²) = -67.02 N m

Therefore, a braking torque of slightly more than 67 N m needs to be applied for one minute (in the opposite direction of rotation, i.e. -67 N m) to bring the turbine to rest in one minute.