If the polar modulus has the same value for both, calculate the diameter of the later and the ratio of their torsional rigidities.

Given:

G for steel= 2.4 x G for alloy

we are given with the following data
for a hollow steel shaft the internal diameter is 200 mm and external diameter is 300 mm and we are given with the rigidity modulus of steel as $2.4\times G\ N\over mm^2$

now whatever to be found out is the dia and rigidity modulus of the solid shaft

The condition that is given to find out the above diameters is the polar modulus for steel =polar modulus of the alloy

$Z {p} [steel]$= $Z{p}[alloy]$

$\pi\times (d{1}^4-d{2}^4\over d{2}$ =$Z{p} \ steel$

$Z{p} [Steel]$ is 2127120$mm^3$

$\pi\times d^3\over 16$ =$Z{p}Alloy$

dia of the alloy is 221.26 mm