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## Aim:

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

## Apparatus:

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

Theory:

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

(1)

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

(2)

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

(3)

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

(4)

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

(5)

## Application:

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