Suppose that a cord is wrapped around the rim a disk of radius $R$. The disk is allowed to rotate around its central axis (the line passing through the center and perpendicular to the disk surface). The force from the cord is $F$. Then I am told that the magnitude of torque on the disk is $RF$.
I could not understand how $RF$ follows from the definition net torque $T= \sum \vec r_i \times \vec F_i$ when the sum is taken over all particle. Things become more confusing as I notice that the force $F_i$ on any single particle of the object must not be zero, because each particle is rotating together with the rigid object.
Any help is appreciated.
Additional Info: The fact that net $T=\sum \vec r_i \times \vec F_i$ is used, for example, in the proof of Newton's second law for rotation $T = I \varepsilon$. The proof (as far as I know) proceeds from the case of a single particle and then generalizes to rigid objects by considering an object as being a combination of many particles.