Pulley - on which side is tension bigger? Two different answers I have the following problem (see picture). The mass of $A$ is 3kg and the mass of $B$ is 2kg. I am told the pulley is not smooth and the difference in tension on either side $\Delta T$ is given by $$\Delta T = 3 + 0.3a$$ where $a$ is the acceleration of the two particles. 
Now I am unsure why the tension on the left has to be bigger. The reason given in the book is that $A$ is more massive. However, if I do the calculation supposing that the tension of the right is greater, I have $$A : 3g - T = 3a$$$$B : T+ 3 + 0.3a - 2g = 2a$$ and combining these I find $a=2.78$ m/s$^2$. This is different to the answer of 1.26 m/s$^2$ we get if we presume the tension on the left is greater. But why do I even get an answer? 
This is part of a bigger issue I am having, as I am trying to apply a similar approach to considering the problem of two masses, one on a table, and one hanging freely, both connected by a string that passes over a non-smooth pulley. However, again, I am not sure on which side of the string the tension will be greatest. 
Any help is thoroughly appreciated. 
 A: The setup of the problem seems to be telling you that the pulley only resists motion via inertia and friction. 
So the unequal gravitational forces on A and B will result in A going down, or the system being stuck. A lighter mass can’t lift a heavier mass without help!
Alternately, if you don’t trust a physical argument, you can assume B is going down. (Be careful with the sign of the “a” term for the pulley.). You’ll find an internally-inconsistent answer, which tells you your assumption about direction was wrong. 
A: Assuming that you have provided all of the information in the problem, both solutions are possible. However, I suspect that you have omitted to say that the system is released from rest.
Case 1 : If the system was released from rest then A would go down, because it is heavier. Friction $f=\Delta T$ in the pulley opposes motion and acts down on the right. Because the pulley is massless the forces on it must balance at all times - otherwise, with a finite resultant torque it would have infinite acceleration. So the tension would be greater on the left : $T_L=f+T_R$. In this case you find that $a=1.26m/s^2$.
Case 2 : However, if the system was not released from rest then B could be moving down initially - eg propelled by an impulse. The friction force then acts towards the left, so tension on the right is greater : $T_L+f = T_R$. In this case you find that $a=2.78m/s^2$. 
In both cases the acceleration is down on the left. In Case 2 block A is accelerating downwards (=decelerating upwards) faster than in Case 1 because friction is acting in the same direction as gravity on A, whereas in the Case 1 friction is opposing gravity on A. In Case 2 the two blocks will eventually change direction, friction will change direction also, and we're back to the situation in Case 1.
