Q.A cylinder of length 1.5 m and diameter 4 cm is fixed at one end. A tangential force of $4\times10^{5}$ N is applied at the other end. If the rigidity modulus of the cylinder is $6\times10^{10}$ N m$^{-2}$ then, calculate the twist produced in the cylinder.

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 a rod, cylinder, or shaft twists under an applied tangential force or torque. Table 7.1 of the chapter lists measured rigidity-modulus values (in units of $10^{10}$ N m$^{-2}$) for common materials: steel (8.0), aluminium (2.5), copper (4.0), iron (6.4), and glass (2.6) -- steel again has the highest value among these, meaning it resists twisting more strongly than any of the other four materials listed. The three elastic moduli of a material are not independent of each other: they are tied together, along with Poisson's ratio $\mu$, by the standard relations $Y=2\eta_R(1+\mu)=3K(1-2\mu)$, which let any one of Y, K, $\eta_R$, or $\mu$ be computed from the other two.