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 \( r_2 \).