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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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$(π μ B b) a$

$(2 π μ B b) Δ a$

$(π μ B b) Δ a$

$(4 π μ B b) Δ a$

answer is D.

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Detailed Solution

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

$V = π a^{2} b$

$△ V = 2 π a b △ a$

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

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

$\Rightarrow N = B \frac{{(2 π a)}^{2} △ a b^{2}}{π a^{2} b}$

Force need to push the cork.

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

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