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 θ, the twist angle at the joint will be

To find the twist angle at the joint of the two rods, we can analyze the torsional behavior of the rods connected in series.

Let the two rods be:
Rod 1 (larger rod): Length $L_1=l$, Radius $r_1=r$.
Rod 2 (smaller rod): Length $L_2=l/2$, Radius $r_2=r/2$.
Both rods are made of the same material, so they have the same shear modulus (modulus of rigidity) $\eta$.

The torsional rigidity (or torque per unit twist) $C$ of a cylindrical rod of length $L$, radius $a$, and shear modulus $\eta$ is given by the formula:
$C=\frac{\pi \eta a^4}{2L}$

Let's find the torsional rigidity of each rod:
For Rod 1 (larger rod):
$C_1=\frac{\pi \eta r^4}{2l}$
For Rod 2 (smaller rod):
$C_2=\frac{\pi \eta {(r/2)}^4}{2(l/2)}=\frac{\pi \eta r^4}{16·l}=\frac{1}{8}\left(\frac{\pi \eta r^4}{2l}\right)=\frac{C_1}{8}$
This gives the relation:
$C_1=8C_2$

The free end of the smaller rod (Rod 2) is fixed to a rigid base. Let the twist angle at the joint be ${\theta }_j$.
Since the small rod is fixed at one end and connected to the joint at the other, the angle of twist across the smaller rod is:
${\theta }_2={\theta }_j-0={\theta }_j$

The free end of the larger rod (Rod 1) is given a twist of $\theta$. Therefore, the angle of twist across the larger rod is:
${\theta }_1=\theta -{\theta }_j$

Since the two rods are joined in series, the restoring torque $\tau$ acting on both rods must be equal in equilibrium:
$\tau =C_1{\theta }_1=C_2{\theta }_2$

Substituting the expressions for ${\theta }_1$, ${\theta }_2$, and the relation between $C_1$ and $C_2$:
$8C_2(\theta -{\theta }_j)=C_2{\theta }_j$

Dividing both sides by $C_2$ ($C_2\ne 0$):
$8(\theta -{\theta }_j)={\theta }_j$
$8\theta -8{\theta }_j={\theta }_j$
$8\theta =9{\theta }_j$
${\theta }_j=\frac{8\theta }{9}$

Thus, the twist angle at the joint is $8\theta /9$.