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(D) The change in volume of the cork is $\Delta V = V_{initial} - V_{final} = \pi (a + \Delta a)^2 b - \pi a^2 b \approx \pi (a^2 + 2a \Delta a) b - \

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$(4\pi \mu Bb) \Delta a$

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(D) The change in volume of the cork is $\Delta V = V_{initial} - V_{final} = \pi (a + \Delta a)^2 b - \pi a^2 b \approx \pi (a^2 + 2a \Delta a) b - \pi a^2 b = 2\pi a b \Delta a$.The volumetric strain is $\frac{\Delta V}{V} = \frac{2\pi a b \Delta a}{\pi a^2 b} = \frac{2 \Delta a}{a}$.The pressure $P$ exerted by the cork on the walls of the bottle is given by $P = B 	imes 	ext{volumetric strain} = B \left( \frac{2 \Delta a}{a} ight)$.The normal force $N$ exerted by the cork on the inner surface of the bottle opening is $N = P 	imes A_{surface} = \left( \frac{2B \Delta a}{a} ight) 	imes (2\pi a b) = 4\pi B b \Delta a$.The frictional force $f$ required to push the cork is $f = \mu N = \mu (4\pi B b \Delta a) = (4\pi \mu B b) \Delta a$.

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