Question

Two parallel and opposite forces, each of magnitude $4000\, N$, are applied tangentially to the upper and lower faces of a cubical metal block $25\, cm$ on a side. Find the displacement of the upper surface relative to the lower surface (in $\times 10^{-5} cm$ ). The shear modulus for the metal is $80\, Gpa$.

Answer 1

We use the approximate from $S=F /(A \phi)$.
Putting $S=8 \times 10^{10} N / m ^{2}, F=4000\, N$ and $A=(0.25\, m )^{2}$
$=6.25 \times 10^{-2} m ^{2}$ in the above equation and solving for $\phi$, we get
$\phi=\frac{(4000 N )}{\left(6.25 \times 10^{-2} m ^{2}\right)\left(8 \times 10^{10} N / m ^{2}\right)}=8.0 \times 10^{-7} rad$
The displacement of the upper surface is given by $d=L \phi$, where $L$ is an edge of the cube.
$\therefore d=\left(8.0 \times 10^{-7}\right)(25\, cm )$
$=2.0 \times 10^{-5} cm$