A bottle has an opening of radius $a$ and length $b$. A cork of length band radius $\left( {a + \Delta a} \right)$ where $\left( {\Delta a < < a} \right)$ is compressed to fit into the opening completely (see figure). If the bulk modulus of cork is $B$ and frictional coefficient between the bottle and cork is $\mu $ then the force needed to push the cork into the bottle is
JEE MAIN 2016, Diffcult
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$Stress = \frac{{Normal\,force}}{{Area}} = \frac{N}{A} = \frac{N}{{\left( {2\pi a} \right)b}}$
$Stress = B \times strain$
$\frac{N}{{\left( {2\pi a} \right)b}} = B\frac{{2\pi a\Delta a \times b}}{{\pi {a^2}b}}$
$ \Rightarrow N = B\frac{{{{\left( {2\pi a} \right)}^2}\Delta a{b^2}}}{{\pi {a^2}b}}$
Force needed to push the cork.
$f = \mu N = \mu 4\pi b\Delta aB = \left( {4\pi \mu Bb} \right)\Delta a$
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