I have encountered some problems where there is different tensions in a massless string. How is this possible? A good example would be
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In a massless string the sum of forces acting on every part of the string must be zero, otherwise, if the force was finite, the string would get an infinite accelleration $a = \lim_{m\rightarrow 0} F/m = \infty$. (The more physical way to say it is that if the mass is very small the force must be very small for finite accelerations). So in frictionless and massless pulleys/cylinders, where there are no additional forces, the tension is the same on both sides. In the case you have presented there is friction between the rope and the cylinder (it is rough enough to not slip). This force accounts for the difference in tension on either side of the cylinder (which must be such that the angular acceleration is compatible with the linear acceleration).
