When to use the formula Txy $=$ T r$/$I and Txy$=$ T r$/$J$?$ Like in the following question $($Example $1)$ we used Txy $=$ T r $/$J why?

$($Example $1)$ The shaft shown is made of an aluminum with the following properties:

Yield stress in tension $=165\frac{N}{m}{m}^2$; Yield stress in compression $=179$

$\frac{N}{m}{m}^2$; Ultimate strength $=310\frac{N}{m}{m}^2$; Elastic modulus$=6\mathrm{.}9x{10}^4$

$\frac{N}{m}{m}^2$; Poisson's ratio $=\frac{1}{3}$

Use the factor of $2$ with respect to failure by slip to determine the maximum

permissible radius of the shaft as predicted by: $($a$)$ The maximum normal

stress theory. $($b$)$ the maximum shear stress theory. $($c$)$ The maximum

distortion energy theory.

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But for the second question $($Example $2)$ we used Txy$=$T r $/$ I

Why?

Can inhave the soltuopn for both

Example $2-[$Example $12\mathrm{.}1($Benham& Armstrong$)]$:

A mild steel shaft of $50mm$ diameter is subjected to a bending

moment of $1\mathrm{.}9$ kNm $.$ If the yield point of steel in simple tension

is $200MN/$ m$2,$ find the maximum torque that can also be

applied according to:

a$)$ Maximum shear stress theory.

b$)$ Maximum shear strain energy $($Von Mises$)$ theory.