A bottle has an opening of radius $\mathrm{a}$ and length $\mathrm{b}$. A cork of length $\mathrm{b}$ and radius $\left(\mathrm{a}+∆\mathrm{a}\right)$ where $\left(∆\mathrm{a}\ll \mathrm{a}\right)$, is compressed to fit into the opening completely (see figure). If the bulk modulus of cork is $\mathrm{B}$ and the coefficient of friction between the bottle and cork is $\mathrm{\mu }$, then the force needed to push the cork into the bottle is