Topic: Strength of Materials
A tapering bar (diameters of end sections being $d_1$ and $d_2$) and a bar of uniform cross-section d have the same length and are subjected to the same axial pull. Both the bars will have same extension if d is equal to
- A. $(d_1+d_2)/2$
- B. $\sqrt{d_1d_2}$
- C. $\sqrt{d_1d_2}/2$
- D. $\sqrt{d_1} + \frac{d_2}{2}$
Answer: B. $\sqrt{d_1d_2}$
Explanation: The extension of a uniformly tapering circular bar is $\Delta L = \frac{4PL}{\pi E d_1 d_2}$.
The extension of a uniform circular bar is $\Delta L = \frac{4PL}{\pi E d^2}$.
Equating both extensions: $\frac{4PL}{\pi E d_1 d_2} = \frac{4PL}{\pi E d^2} \Rightarrow d^2 = d_1 d_2 \Rightarrow d = \sqrt{d_1 d_2}$.