rotation~

I will solve this question in a general way, ie, everything symbolic, and then plug the numbers back.

Let m1 be 2kg mass,
m2 be 4kg mass,
M be mass of the pulley = 2 kg,
T1 be tension on m1,
T2 be tension on m2,
TauF be frictional torque = 0.5 Nm,
Tau be the net torque on the pulley,
R be radius of the pulley = 6 cm,
I be moment of iniertia of the pulley
a be the acceleration,
Alpha be the angular acceleration of the pulley,
h be the distance from ground of m2 = 1 m

Step 1: write down the three equations for m1, m2 and M.

The equations for m1 and m2 is obviously given by:
T1 - m1 g = m1 a ------------------------(1)
m2 g - T2 = m2 a ------------------------(2)
For the pulley, clockwise torque = T2 R
anti-clockwise torque = T1 R + TauF
So, net torque is given by: Tau = (T2 - T1)R - TauF
Since Tau = I alpha, so , we have the equation of the pulley given by:
(T2 - T1)R - TauF = I alpha
(T2 - T1)R^2 - TauF R = I a ----------------------(3) because a = R alpha

Don't be confused by all the above symbols. Keep calm. Among all the symbols in the three equations, only T1, T2 and a are unknowns. All other are known constants. Now we have three linear equations in three unkowns. We can solve them.

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Step 2: Solve the 3 equations in 3 unknowns.

This step is simply f.3 algebra and so I will not show the steps explicitly. The result for acceleration, a, is given by:
a=(m2 g R^2 - m1 g R^2 - TauF R)/(m2 R^2 + m1 R^2 +I)
=(m2 g R^2 - m1 g R^2 - TauF R)/(m2 R^2 + m1 R^2 + 0.5 M R^2)
because the moment of iniertia for a circular disc is I = 0.5 M R^2.
So,
a = (m2 g - m1 g - TauF/R)/(m2 + m1 + 0.5 M) -------------------(4)

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Step 3: Find out the time needed for m2 to reach ground.

Using the equations of motion for constant acceleration, h = ut + 0.5 at^2, and that u =0 as the masses are released from rest, we have the result:
t = sqrt(2 h / a), where sqrt means square root. Plug the a obtained from step 2, we get:
t = sqrt[ 2h (m1 + m2 + M/2) / (m2 g - m1 g - TauF/R) ]

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Step 4: Plug back numbers.

In fact, I myself would regard the above as answer. But if you need numbers, plug it back.
t = 1.11 sec. I used g=9.8 ms^-2

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