Two circular shafts made of same material, one solid $$(S)$$ and one hollow $$(H)$$, have the same length and polar moment of inertia. Both are subjected to same torque. Here. $${\theta _S}$$ is the twist and $${\tau _S}$$ is the maximum shear stress in the solid shaft, whereas $${\theta _H}$$ is the twist and $${\tau _H}$$ is the maximum shear stress in the hollow shaft. Which one of the following is TRUE?

A hollow shaft of $$1$$ m length is designed to transmit a power of $$30$$ kW at $$700$$ rpm. The maximum permissible angle of twist in the shaft is $${1^0}.$$ The inner diameter of the shaft is $$0.7$$ times the outer diameter. The modulus of rigidity is $$80$$ GPa. The outside diameter (in mm) of the shaft is _______

A solid circular shaft needs to be designed to transmit a torque of 50N.m. If the allowable shear stress of the material is $$140$$MPa, assuming a factor of safety of $$2,$$ the minimum allowable design diameter in mm is

A torque $$T$$ is applied at the free end of a stepped rod of circular cross-sections as shown in the figure. The shear modulus of the material of the rod is $$G.$$ The expressions for diameter to produce an angular twist $$\theta $$ at the free end is