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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 frictional coefficient between the bottle and cork is $μ$ then the force needed to push the cork into the bottle is:

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$(2 π μ B b) Δ a$

$(4 π μ B b) Δ a$

$(π μ B b) Δ a$

$(π μ B b) a$

answer is D.

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Detailed Solution

$S t r e s s = \frac{N o r m a l f o r c e}{A r e a} = \frac{N}{A} = \frac{N}{(2 π a) b}$

$V = π a^{2} b, Δ V = 2 π a b Δ a$

$S t r e s s = B \times s t r a i n$

$\frac{N}{(2 π a) b} = B \frac{2 π a Δ a \times b}{π a^{2} b} [∵ S t r a i n = \frac{Δ V}{V}] \Rightarrow N = B \frac{{(2 π a)}^{2} Δ a b^{2}}{π a^{2} b}$

Force need to push the cork. $f = μ N = \frac{μ \times B \times 4 π^{2} a^{2} b^{2} Δ a}{π a^{2} b} = 4 (π μ B b) Δ a$

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