Question 215: Properties of Matter

Question

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 $	heta_2=2 	heta_1$. If the shear moduli of material 1 is $4 	imes 10^9 \mathrm{~N} / \mathrm{m}^2$, then shear moduli of material 2 is $x 	imes 10^9 \mathrm{~N} / \mathrm{m}^2$, where value of $x$ is ________.

Accepted Answer

$$\begin{aligned}
&	ext { Deformation angle }\
&\begin{aligned}
& 2 	heta_1=	heta_2 \
& \Rightarrow 2 \frac{\sigma_1}{\eta_1}=\frac{\sigma_2}{\eta_2}
\end{aligned}
\end{aligned}$$
$$\begin{aligned}
& \Rightarrow 2\left(\frac{F}{\ell d_1 \eta_1}ight)=\frac{F}{\ell d_2 \eta_2} \
& \Rightarrow \eta_2=\frac{\eta_1}{4}=1 	imes 10^9 \Rightarrow x=1
\end{aligned}$$

Properties of Matter

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