How Long Does It Take for m1 to Reach the Floor in an Atwood Machine?

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To determine the time it takes for m1 to reach the floor in an Atwood machine, the system involves two masses, m1 and m2, with m1 starting at a height of 4.70 m. The relevant equations include conservation of mechanical energy and the relationship between torque and angular acceleration. The attempt to solve the problem involved calculating the final angular velocity and inertia of the pulley but encountered difficulties in finding angular acceleration and time. A suggestion was made to draw a free-body diagram to clarify the forces at play. The discussion emphasizes the importance of correctly applying physics principles to solve the problem.
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Homework Statement



Consider the system shown in the image attached with m1 = 27.0 kg, m2 = 13.6 kg, R = 0.290 m, and the mass of the pulley M = 5.00 kg. Object m2 is resting on the floor, and object m1 is 4.70 m above the floor when it is released from rest. The pulley axis is frictionless. The cord is light, does not stretch, and does not slip on the pulley.

Calculate the time interval required for m1 to hit the floor.



Homework Equations


6be716e191ceca92e0712ac4b3254a8c.png

I*Angular Accleration=torque
Conservation of Mechanical Energy( K+U, final=K+U, initial)



The Attempt at a Solution


I played around with the formulas starting with the mechanical energy formula. I got this:

omega [final] = (1/R) [(2*(m2-m1)gh)/(m1+m2+(I/R^2))]^(1/2)
I=.5MR^2, M=mass of pulley

So I plugged in the values and got omega[final]= 119

Now, I got to find time, but I'm stuck. I plug in 119 in the formala:

Omega[final] = omega[initial] +angular acceleration*time, omega[initial]= 0

I have to determine the angular accleration (equals torque/Inertia)

Inertia equals to .21075 (.5*5kg*.29m^2). To find torque, I could use Tau=rFSin(theta), but I end trying to find Force and theta. I don't really know where to go from here...
 
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Draw a free-body diagram first.

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