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Rigidity Modulus -Static Torsion





To determine the rigidity modulus of the material of a given cylindrical rod through telescope and scale method.




Searle's static torsion apparatus: rod with attached pulley, weight hanger, slotted weights, telescope, mirror and scale.




Shear modulus, or rigidity modulus n is defined as the ratio of stress F/A to strain Δx/l when a shearing force F is applied to a rigid block of height l and area A. Δx is the deformation of the block, and




This is similar to what happens when a torque τ is applied to a rigid rod of length l and radius r. Looking at the cross-section of the rod, consider a ring of width dr' at radius r' , which will have area 2πr'dr', with force applied tangentially. The weighted average force over the cross-sectional area A of the rod is then



If the torque deforms the rod by twisting it through a small angle θ, the deformation distance (corresponding to Δx) at the outside edge of the rod is approximately  θr. The definition of the rigidity modulus n becomes     



In our apparatus the torque τ is supplied by hanging a weight of mass M from a string wound round a pulley of radius R, so τ =MgR and our definition of rigidity modulus n becomes






Now suppose we mount a small mirror on the rod at distance l from its fixed end, and look at a centimeter scale in the mirror through an adjacent telescope, both at distance D from the mirror. When the rod deforms and the mirror rotates through a small angle θ  , we look at a point on the scale a distance approximately S=2Dθ from the original point, which was aligned with the telescope. We can measure D and S and substitute θ =S/2D in our definition of rigidity modulus n, to get






Engineers consider the value of shear modulus when selecting materials for shafts, which are rods that are subjected to twisting torques.



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