Question

A hollow steel shaft of external diameter 100 mm and internal diameter 50 mm is to be replaced by a solid alloy shaft. Assuming the same value of polar modulus for both, the diameter of the solid alloy shaft will be

• A. $10\times 3\sqrt{9375}mm$
• B. $10\times 2\sqrt{9375}mm$
• C. $10\times 3\sqrt{\left(\frac{9375}{10}\right)}mm$
• D. $3\sqrt{9375}mm$

Answer 1

The correct option is C $10\times 3\sqrt{\left(\frac{9375}{10}\right)}mm$

Polar modulus is also called polar section modulus or torsional section modulus

$\frac{I_P}{R}=Polarmodulus$

$\therefore \frac{I_{P_1}}{R_1}=\frac{I_{P_2}}{R_2}$

$\Rightarrow \frac{\left(\pi /32\right)\times (D_o^4-D_i^4)}{D_o/2}=\frac{\left(\pi /32\right)\times D^4}{D/2}$

$\Rightarrow D_o^3[1-{\left(\frac{D_i}{D_o}\right)}^4]=D^3$

$\Rightarrow {100}^3[1-{\left(\frac{50}{100}\right)}^4]=D^3$

$\Rightarrow D=100\times 3\sqrt{0.9375}$

$\Rightarrow D=10\times 3\sqrt{\frac{9375}{10}}$