Mechanical Properties of Solids

JEE Main level, single correct, medium

101 of 118

Question 101 of 118

A bottle has an opening of radius aa and length bb. A cork of length bb and radius (a+∆a)\left(a+∆a\right) where (∆a≪a)\left(∆a≪a\right), is compressed to fit into the opening completely (see figure). If the bulk modulus of cork is BB and the coefficient of friction between the bottle and cork is μ\mu, then the force needed to push the cork into the bottle is

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Answer: (D) (4πμBb)∆a\left(4\pi \mu Bb\right)∆a

Pressure=NA=N(2πa)bPressure=\dfrac{N}{A}=\dfrac{N}{\left(2\pi a\right)b}

⇒stress=B×strain\Rightarrow stress=B\times strain

N(2πa)b=B2πa∆a×bπa2b\dfrac{N}{\left(2\pi a\right)b}=B\dfrac{2\pi a∆a\times b}{{\pi a}^{2}b}

⇒N=B(4πb△a)\Rightarrow N=B\left(4\pi b△a\right)

f=μN=μ4πbBΔaf=\mu N=\mu 4\pi bB\Delta a

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