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A bottle has an opening of radius a and length b. A cork of length b and radius (a + $Δ$ a) where ( $Δ$ a $≪$ a), is compressed to fit into the opening completely (see figure), If the bulk modulus of cork is B and the coefficient of friction between the bottle and cork is $μ$ , then the force needed to push the cork into the bottle is

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$(4 π μ B b) Δ a$

$(2 π μ B b) Δ a$

$(π μ B b) Δ a$

answer is D.

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Detailed Solution

$(4 π μ B b) Δ a$ Pressure = $\frac{N}{A} = \frac{N}{(2 π a) b}$ $\Rightarrow$ Stress = B $\times$ Strain

$\frac{N}{(2 π a) b} = B \frac{2 π a Δ a \times b}{π a^{2} b}$ $\Rightarrow N = B f = μ N = μ 4 π b B Δ a .$

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