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 :
\[\Delta V\simeq -2\pi ab\Delta a\] \[{{V}_{f}}=\pi {{a}^{2}}b\] \[\frac{\Delta V}{V}=\frac{-2\pi ab\Delta a}{\pi {{a}^{2}}b}=\frac{-2\Delta a}{a}\] \[\Rightarrow \,\Delta P=\frac{2\beta \Delta a}{a}\] Normal force = \[=\frac{2\beta \Delta a}{a}2\pi ab\] \[=4\pi \beta \,\,b\,\Delta a\] friction \[=\mu N\] \[=4\pi \mu \beta \,\,b\,\,a\]
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