Compute the ratio of the torque:

Compute the ratio of the torque transmitted by a hollow & a solid shaft of the similar material, length & weight.

Solution

d_i = internal diameter of hollow shaft

 d₀ = external diameter of hollow shaft

d = diameter of solid shaft

Similar material means modulus of rigidity and density is similar. The weight being equal means the cross-sectional areas of solid & hollow shafts are similar:

i.e.

where

∴  n = d_i / d_o

d = do

τs = 16Ts / π d ³ =  16Ts / π d ³ (1 - n² ) ^(3/2)

For same material τ_s = τ_h

 ∴         Ts    /(1 - n² ) ^(3/2)   =Th/(1 - n⁴ )²