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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 7

  • question_answer
    A bottle has an opening of radius a and length b. A cork of length b and radius \[(a+\Delta a)\]where\[(\Delta a<<a)\] 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 then the force needed to push the cork into the bottle is           

    A)  \[(\pi \mu Bb)a\]

    B)  \[(2\pi \mu Bb)\Delta a\]

    C)  \[(\pi \mu Bb)\Delta a\]  

    D)  \[(4\pi \mu Bb)\Delta a\]

    Correct Answer: D

    Solution :

    [d]: Bulk modulus, \[\text{B}=\frac{\text{Normal}\,\text{stress}}{\text{Volumetric}\,\text{strain}}\] \[P=\frac{N}{A}=\frac{N}{(2\pi a)b}\] Volumetric strain \[=\frac{2\pi a\Delta a\times b}{\pi {{a}^{2}}b}=\frac{2\Delta a}{a}\] \[\therefore \]\[B=\frac{N}{2\pi ab}\times \frac{a}{2\Delta a}\]\[N=4\pi b\Delta a\times B\] \[\therefore \]Required force = Frictional force \[=\mu N=(4\pi \mu Bb)\Delta a\]


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