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

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Answer & Solution

Correct Answer

(D) \(\left( {4\pi \mu Bb} \right)\,\Delta a\)

Detailed explanation

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

\(Stress = B \times strain\)

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

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

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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