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A rod of length $l$ and radius $r$ is joined to a rod of length $l / 2$ and radius $r / 2$ of same material. The free end of small rod is fixed to a rigid base and the free end of larger rod is given a twist of $\theta^{\circ}$, the twist angle at the joint will be
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$8 \theta / 9$
$\tau=C . \theta=\frac{\pi w^4 \theta}{2 L}=$ Constant
$\Rightarrow \frac{\pi \eta r^4\left(\theta-\theta_0\right)}{2 l}=\frac{\pi \eta(r / 2)^4\left(\theta_0-\theta^{\prime}\right)}{2(l / 2)}$
$\Rightarrow \frac{\left(\theta-\theta_0\right)}{2}=\frac{\theta_0}{16} \Rightarrow \theta_0=\frac{8}{9} \theta$
$\Rightarrow \frac{\pi \eta r^4\left(\theta-\theta_0\right)}{2 l}=\frac{\pi \eta(r / 2)^4\left(\theta_0-\theta^{\prime}\right)}{2(l / 2)}$
$\Rightarrow \frac{\left(\theta-\theta_0\right)}{2}=\frac{\theta_0}{16} \Rightarrow \theta_0=\frac{8}{9} \theta$
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