Two slabs with square cross section of different materials (1, 2) with equal sides ($$l$$) and thickness $$d_1$$ and $$d_2$$ such that $$d_2 = 2d_1$$ and $$l \gt d_2$$. Considering lower edges of these slabs are fixed to the floor, we apply equal shearing force on the narrow faces. The angle of deformation is $$\theta_2 = 2\theta_1$$. If the shear moduli of material 1 is $$4 \times 10^9$$ N/m$$^2$$, then shear moduli of material 2 is $$x \times 10^9$$, where value of x is ________.

For a slab under a tangential (shearing) force $$F$$,
 • the shearing (tangential) stress is $$\tau = \dfrac{F}{A}$$, where $$A$$ is the area of the face on which the force acts.
 • the shearing strain is $$\gamma = \tan\theta \approx \theta$$ for small angles.
 • shear modulus (rigidity modulus) is defined by $$G = \dfrac{\text{shear stress}}{\text{shear strain}} = \dfrac{\tau}{\gamma}.$$

Both slabs have square cross-section of side $$l$$ and thickness $$d$$. The force $$F$$ is applied on a narrow face of each slab; that face has dimensions $$l \times d$$, so its area is $$A = l\,d$$.

Thickness $$d_1$$, narrow-face area $$A_1 = l\,d_1$$. Shear stress $$\tau_1 = \dfrac{F}{A_1} = \dfrac{F}{l\,d_1}$$. Let the small angle of deformation be $$\theta_1$$, so shear strain $$\gamma_1 = \theta_1$$.

Hence $$G_1 = \dfrac{\tau_1}{\gamma_1} = \dfrac{F}{l\,d_1\,\theta_1} \quad -(1)$$.

Thickness $$d_2 = 2d_1$$, narrow-face area $$A_2 = l\,d_2 = l\,(2d_1) = 2l\,d_1$$. Shear stress $$\tau_2 = \dfrac{F}{A_2} = \dfrac{F}{2l\,d_1} = \dfrac{\tau_1}{2}$$. Given angle $$\theta_2 = 2\theta_1$$, so shear strain $$\gamma_2 = \theta_2 = 2\theta_1$$.

Therefore $$G_2 = \dfrac{\tau_2}{\gamma_2} = \dfrac{\tau_1/2}{2\theta_1} = \dfrac{\tau_1}{4\theta_1} \quad -(2)$$.

Divide $$(2)$$ by $$(1)$$:

$$\dfrac{G_2}{G_1} = \dfrac{\tau_1/(4\theta_1)}{F/(l\,d_1\,\theta_1)} = \dfrac{F}{4l\,d_1\,\theta_1}\times\dfrac{l\,d_1\,\theta_1}{F} = \dfrac{1}{4}.$$

Hence $$G_2 = \dfrac{G_1}{4}$$.

Given $$G_1 = 4 \times 10^9 \text{ N m}^{-2}$$,

$$G_2 = \dfrac{4 \times 10^9}{4} = 1 \times 10^9 \text{ N m}^{-2}.$$

The required value is $$x = 1$$.