Q.A metal cube of side 0.20 m is subjected to a shearing force of 4000 N. The top surface is displaced through 0.50 cm with respect to the bottom. Calculate the shear modulus of elasticity of the metal.

The RIGIDITY MODULUS (or SHEAR MODULUS), $\eta_R$, is one of the three elastic moduli of a material, defined as the ratio of shearing stress to shearing strain: $\eta_R=\dfrac{\sigma_s}{\varepsilon_s}=\dfrac{F/A}{x/h}$, where F is the tangential force applied over area A, and x/h is the resulting shear (the tangent of the angle of shear, which for a small angle equals the angle itself, $\theta$). Its SI unit is N m$^{-2}$ (pascal), matching the other elastic moduli, since strain is dimensionless. Physically, the rigidity modulus measures how strongly a material resists being TWISTED or sheared: a SMALL $\eta_R$ means a material twists easily for a given applied torque, since the restoring torque $\tau$ developed in a twisted wire or rod is proportional to the angle of twist $\theta$ ($\tau\propto\theta$), and a small rigidity modulus means a large twist angle results from even a modest torque. TWIST OF A SOLID CYLINDRICAL ROD. For a solid cylindrical rod of length L and radius r, rigidly fixed at one end, applying a tangential force F at the rim of the free end (producing a torque $\tau=Fr$ about the rod's axis) twists that free end through an angle $\theta=\dfrac{2\tau L}{\pi r^4\eta_R}$ -- equivalently, the couple needed per unit angle of twist (the torsional rigidity) of the rod is $C=\dfrac{\pi\eta_R r^4}{2L}$. This torsion relation extends the simple cube-shear picture of rigidity modulus to the practically important case of a shaft or rod twisted about its own axis, and is the formula used whenever a problem asks for the angle through which …