why friction is
not considered in fbd of pulley while it produce torque in one
direction as it has tendency to rotate as you have drawn the
figure showing friction in figure
if you draw free body diagram of pully only without string then
we have to show friction, but if you draw fbd of string + pulley
as a system then you cannot show friction as it act on string
also in opposite direction. Moreover , fR=(T2-T1)R. static
friction is the cause of difference in tension