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QJEEM16Phee47: JEE Main 2016 Physics - Class 11 JEEM16Phee47 Solution: A bottle has an opening of radius a and length b. A cork of…

A bottle has an opening of radius a and length b. A cork of length b and radius (a + Δ\Delta a) where ( Δ\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 μ\mu then the force needed to push the cork into the bottle is :

JEE Main 2016 (Online) 10th April Morning Slot Physics - Properties of Matter Question 226 English

Multiple Choice Options

  • A ( π\pi μ\mu B b) Δ\Delta a
  • B (2 π\pi μ\mu B b) Δ\Delta a
  • C ( π\pi μ\mu B b) a
  • D (4 π\pi μ\mu B b) Δ\Delta a

Step-by-Step Verified Solution

Bulk modulus, B  =   ΔPΔVV{{\Delta P} \over {{{\Delta V} \over V}}}

Vi  =   π\pi (a + Δ\Delta a)2b

Vf  =   π\pi a2b

∴\therefore     Δ\Delta V = 2 π\pi ab Δ\Delta a

∴\therefore     ΔVV{{{\Delta V} \over V}} = 2πabΔaπa2b{{2\pi ab\Delta a} \over {\pi {a^2}b}} = 2Δaa{{2\Delta a} \over a}

∴\therefore     Δ\Delta P = B ×\times ΔVV{{{\Delta V} \over V}}

      = B ×\times 2Δaa{{2\Delta a} \over a}

Normal Force (N) = Δ\Delta P ×\times A

       = B ×\times 2Δaa{{2\Delta a} \over a} ×\times (2 π\pi a)b

       =   4 π\pi B Δ\Delta ab

∴\therefore    Frictional force = μ\mu N = (4 π\pi μ\mu Bb) Δ\Delta a

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