Using Forces and Torques to solve a pulley problem

[kepherax]

Homework Statement

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Relevant Equations

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I am able to solve this easily using energy conservation, but I'm curious how one would go about solving this using forces and torques?

 

[Delta2]

I see you have already make one equation with the net torque on the pulley. You can make two more equations by applying Newton's 2nd law on each of the masses. Then one more equation by the fact that the pulley rotates without slipping. So in total you ll have 4 equations with 4 unknowns (The two tensions, the (common) acceleration of the masses and the angular acceleration of the pulley).

 

[kuruman]

Alternatively, you can use mechanical energy conservation and get the answer with a single equation.