Question

Obtain an expression for the modulus of rigidity. State SI unit and dimensions of 'η'.

Answer 1

modulus of rigidity or shears modulus is the material properties , it is the ratio of shears stress to shears strain. shears stress is the ratio of tension per unit area while shears strain is the ratio of change in length to unit original length.

so, modulus of rigidity or shears modulus, $\eta=\frac{\textbf{shears stress}}{\textbf{shears strain}}$

$\eta=\frac{\frac{F}{A}}{\frac{\Delta{L}}{L}}$

$\eta=\frac{FL}{A\Delta{L}}$ it is the expression for the Modulus of rigidity.

now, S.I unit of $\eta$ is N/m² or Pascal
dimension of $\eta$ is $[ML^{-1}T^{-2}]$

Answer 2

HEY DEAR ... ✌️

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=) Within the elastic limit , upto which Hooke's law is applicable , the ratio of the shearing stress to the shearing strain is called the 'modulus of rigidity' of the material of the body . It is denoted by η .

Let us consider a cube ABCDHIJK ( in figure ) whose lower surface is fixed . When a tangential force F is applied at its upper surface ABIH , then all the layers parallel to ABIH are displaced in the direction of the force . The displacement of a layer is proportional to its distance from the fixed surface . The cube thus takes the new form A'B' CDH'I'JK . The angle ϴ through which the line AD or BC , initially perpendicular to the fixed surface , is turned is called the 'angle of shear' or 'shearing strain' . If ϴ is small , then

ϴ = tan ϴ = BB' / BC

= displacement of upper surface / displacement of upper surface from fixed surface

If A be the area of the upper surface (ABIH) , then

Shearing stress = F / A

.-. modulus of rigidity of the material of cube is

η = F/A / ϴ

η = F / Aϴ

The SI unit of η is 'newton/metere^2' , [Nm^-2] or 'pascal' (Pa) and it's dimensional formula is [ML^-1T^2] .

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HOPE , IT HELPS ... ✌️