Torsional rigidity of a tapering wire:
Let us consider a tapering wire of length l and rigidity of modulus \eta . The end radii are r_1 and r_2 . Here r_1<r_2 as shown in the figures. Let us consider an elementary small spherical disc of radius r and thickness dx at a distance x from the fixed end of radius r_1 of the tapering wire.
From Fig.2 we can write,
\frac{r_2-r_1}{l}=\frac{\delta{r}}{x} \\or,\ \delta{r}=\frac{x}{l}(r_2-r_1)
Again, r=r_1+\delta{r}\\=r_1+\frac{x}{l}(r_2-r_1)
Let, \tau_x be the torsional rigidity i.e., twisting couple per unit twist of the elementary spherical disc, and d\theta be the angle of twist between the two ends of that element, then the applied torque is given by,
\displaystyle{d\Gamma=\tau_x\cdot{d\theta}\\=\frac{\pi\eta{r^4}}{2dx}d\theta}
[Read In Detail]
or, \displaystyle{\frac{d\theta}{d\Gamma}=\frac{2dx}{\pi\eta{r^4}}\\=\frac{2dx}{\pi\eta}\frac{1}{{[r_1+\frac{x}{l}(r_2-r_1)]}^4}}
[putting the value of r ]
Integrating both side we get,
\displaystyle{\frac{\theta}{\Gamma}=\frac{2}{\pi\eta}\int_0^l\frac{dx}{{[r_1+\frac{x}{l}(r_2-r_1)]}^4}}
or, \displaystyle{\frac{\theta}{\Gamma}=\frac{2}{\pi\eta}\int_{r_1}^{r_2}\frac{1}{z^4}\frac{l}{r_2-r_1}dz\\=\frac{2l}{\pi\eta(r_2-r_1)}\int_{r_1}^{r_2}\frac{dz}{z^4}\\=\frac{2l}{\pi\eta(r_2-r_1)}{(\frac{z^{-3}}{-3})}_{r_1}^{r_2}\\=-\frac{2l}{3\pi\eta(r_2-r_1)}(\frac{1}{{r_2}^3}-\frac{1}{{r_1}^3}) \\=\frac{2l}{3\pi\eta(r_2-r_1)}(\frac{1}{{r_1}^3}-\frac{1}{{r_2}^3})\\=\frac{2l}{3\pi\eta}\frac{{r_1}^2+{r_1}{r_2}+{r_2}^2}{{r_1}^3{r_2}^3} }
Let, \displaystyle{ z=r_1+\frac{x}{l}(r_2-r_1)\\or,\ dz=\frac{(r_2-r_1)}{l}dx\\or,\ dx=\frac{l}{(r_2-r_1)}dz }
Or, \displaystyle{\frac{\Gamma}{\theta}=\frac{3}{2}\frac{\pi\eta{r_1}^3{r_2}^3}{l({r_1}^2+{r_1}{r_2}+{r_2}^2)}}
This is the torsional rigidity of a tapering wire of length l and end radii r_1 and r_2 .