question_answer

A bottle has an opening of radius a and length b. A cork of length b and 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 ONLINE 10-04-2016]

A)  \[\left( 2\pi g\mu B\,\,b \right)\Delta a\]

B)       \[\left( \pi \mu B\,\,b \right)\Delta a\]

C)  \[\left( \pi \mu \,B\,\,b \right)a\]   

D)       \[\left( 4\pi \mu \,B\,\,b \right)\Delta a\]

Correct Answer: D Solution : [d] \[\beta \frac{\Delta V}{V}=-\Delta P\]

\[\Delta V\simeq -2\pi ab\Delta a\]

\[{{V}_{f}}=\pi {{a}^{2}}b\]

\[\frac{\Delta V}{V}=\frac{-2\pi ab\Delta a}{\pi {{a}^{2}}b}=\frac{-2\Delta a}{a}\]

\[\Rightarrow \,\Delta P=\frac{2\beta \Delta a}{a}\]

Normal force = \[=\frac{2\beta \Delta a}{a}2\pi ab\]

\[=4\pi \beta \,\,b\,\Delta a\]

friction \[=\mu N\]

\[=4\pi \mu \beta \,\,b\,\,a\]