Question

Two shafts of solid circular cross-section identical except their diameters are subjected to the same torque (T). What will be the ratio of strain energies (U1/U2) stored in both shafts if the diameter of the first shaft is d1 and diameter of the second shaft is d2, respectively?

• A. $\left(\frac{d_1}{d_2}\right)^4$
• B. $\left(\frac{d_1}{d_2}\right)^2$
• C. $\left(\frac{d_2}{d_1}\right)^2$
• D. $\left(\frac{d_2}{d_1}\right)^4$

Hint:

For strain energy in shafts: - Use $U \propto \frac{T^2}{J}$. - The strain energy ratio is proportional to the fourth power of the inverse diameter ratio.

Answer 1

The Correct Option is D

The strain energy ($U$) stored in a shaft subjected to torque is proportional to:
\[U \propto \frac{T^2}{GJ}\]
where:
$T$: Applied torque,
$G$: Shear modulus,
$J = \frac{\pi d^4}{32}$: Polar moment of inertia.
For identical torque and material:
\[\frac{U_1}{U_2} = \frac{J_2}{J_1} = \left(\frac{d_2}{d_1}\right)^4.\]