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The total force acting on the pulley is zero so:

F=mg+T1+T2 (1)Analyzing the torque and angular acceleration about the actual axis of rotation, the axle of the pulley, gives:

τnet=T1R−T2R=Iα (2)If we analyze about point P, the right edge of the pulley where T1 is applied, we get:

τnet=(F−mg)R−T2×2R=(I+mR2)α WRONG(3)Using Equation 1 to eliminate F−mg from Equation 3 gives:

τnet=T1R−T2R=(I+mR2)α WRONG(4)The net torque in Equations 2 and 4 is the same, but the moments of inertia are different so the angular accelerations are also different. Note that if we think of point P as attached to the right-hand string, if

*T1 ≠ T2*then it is accelerating.My question is if we think P as point fixed in space and not attached to to the right-hand string, then what will be the equation of torque about point P?

In the case of instantaneous axis of rotation, we say that the center of rotation is a point in space and does not undergo radial acceleration. So we think of it as an inertial frame of reference. So can't we analyze the torque about point P thinking that it not subject to any kind of acceleration.In that case we should get the actual torque about point P.