Two slabs with square cross section of different materials (1, 2) with equal sides () and thickness and such that and . Considering lower edges of these slabs are fixed to the floor, we apply equal shearing force on the narrow faces. The angle of defor

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Solution

Q.
Two slabs with square cross section of different materials (1, 2) with equal sides ( $l$ ) and thickness $d_{1}$ and $d_{2}$ such that $d_{2} = 2 d_{1}$ and $l > d_{2}$ . Considering lower edges of these slabs are fixed to the floor, we apply equal shearing force on the narrow faces. The angle of deformation is $θ_{2} = 2 θ_{1}$ . If the shear moduli of material 1 is $4 \times 10^{9} N / m^{2}$ , then shear moduli of material 2 is $x \times 10^{9} N / m^{2}$ , where value of x is __________. [2025]

Ans.
(1)

Deformation angle

$2 θ_{1} = θ_{2} \Rightarrow 2 \frac{σ_{1}}{η_{1}} = \frac{σ_{2}}{η_{2}}$

$\Rightarrow 2 (\frac{F}{A_{1} η_{1}}) = \frac{F}{A_{2} η_{2}}$

$\Rightarrow 2 (\frac{F}{l d_{1} η_{1}}) = \frac{F}{l d_{2} η_{2}}$

$\Rightarrow η_{2} = \frac{η_{1}}{4} = 1 \times 10^{9} \Rightarrow x = 1$

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