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Imagine that we want to get a bucket filled with water out of a well whose mechanisms looks something like that I have drawn. Then, we want to calculate the acceleration the bucket would have if we stopped applying torque. The force that would cause the bucket to accelerate would be the tension of the rope.
My question is, can we assume that the force of gravity acting on the bucket is very close to the tension of the rope and use its value in our calculations instead?
I'm not sure I understood the question but if the pulley is massless than T is zero when the torque is zero. IF it has a mass than we need its radius $R$ and its angular mass $I$. So we can write the kinematic relation between the bucket and the angular acceleration and velocity of the pulley $v=\omega R$, $a=\dot\omega R$.
Then we need the energy relation $W={dK\over dt}$ where $W=mgv$ is the power of the weight and $K=\frac12mv^2+\frac12I\omega^2$ is the kinetic energy of the system:
$mgv=mva+I\omega\dot\omega\Rightarrow mgv=mva+IR^2va\Rightarrow a=\frac{mg}{m+IR^2}$
Now you can find $T$ using Newton law $mg-T=am\Rightarrow T=mg\frac{IR^2}{m+IR^2}$
