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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$\tau = C.\theta = \frac{\pi \eta r^{4} \theta}{2L} = Constant$
$\Rightarrow \frac{\pi \eta r^{4} (\theta - \theta_{0})}{2l} = \frac{\pi \eta (r/2)^{4} (\theta_{0} - \theta')}{2(1/2)}$
$\Rightarrow \frac{(\theta - \theta_0)}{2} = \frac{\theta_0}{16} \Rightarrow \theta_0 = \frac{8}{9} \theta$
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