Static analysis of a lead screw and disc friction

I'm doing a fixed study of a quadrate thread lead screw. I deliberation this tract gives a bully explanation of nan problem.

Screw threads statics

To cipher nan infinitesimal connected nan thread nan mean radius of nan thread is used.

$$M = W \space R \space tan(phi_s + alpha)$$

Where M is nan infinitesimal required to raise nan screw to impending motion, W is nan unit load connected nan screw, R is nan mean radius of nan screw, phi_s is nan screw clash angle, and alpha is nan screw thread pitch.

Further down connected nan aforesaid tract a disc clash problem is looked at.

Disc Friction Statics

Here nan infinitesimal is calculated by integrating pinch respect to nan radius.

Adapting this attack to analyse nan screw thread, nan infinitesimal required to raise nan screw would be.

$$M = \frac{2}{3} \frac{(R_o^3 - R_i^3)}{(R_o^2 - R_i^2)} W \space tan(phi_s + alpha)$$

Where R_o is nan extracurricular radius and R_i is nan wrong radius.

These output akin but different results. My mobility is, is nan first attack utilizing nan mean radius simply a simplification/approximation of nan disc clash method of integrating complete radius and nan later will beryllium much accurate? Or is it incorrect to merge complete nan radius erstwhile looking astatine a screw thread analysis? If so, tin you please elaborate connected why.