d
\(Stress = \frac{{Normal\,force}}{{Area}} = \frac{N}{A} = \frac{N}{{\left( {2\pi a} \right)b}}\)

\(Stress = B \times strain\)

\(\frac{N}{{\left( {2\pi a} \right)b}} = B\frac{{2\pi a\Delta a \times b}}{{\pi {a^2}b}}\)

\( \Rightarrow N = B\frac{{{{\left( {2\pi a} \right)}^2}\Delta a{b^2}}}{{\pi {a^2}b}}\)

Force needed to push the cork.

\(f = \mu N = \mu 4\pi b\Delta aB = \left( {4\pi \mu Bb} \right)\Delta a\)