(Solved): SECTION B Answer one question only. 7 A cylindrical cork, with density , length l and radius a, p ...

SECTION B Answer one question only. 7 A cylindrical cork, with density $?$, length $l$ and radius $a$, protrudes a distance $x$ from the neck of a bottle containing a gas at pressure $P_g$ above atmospheric pressure, as shown below. The portion of the cork in the bottle is compressed, and pushes radially outwards on the bottle neck with a constant pressure $P_c$. Show that the cork will slide outwards if $P_g>\frac{2(l?x)?}{a}{P}_c$ where $?$ is the coefficient of friction between the cork and the glass. Assume $?$ takes the same value whether or not the cork is slipping. The cork starts flush with the bottle $(x=0)$, but the pressure in the bottle is just enough for the cork to start slipping forwards. Assuming the pressure in the bottle is constant, and the cork starts with a small outward velocity ${?}_0$ at $t=0$, find the equation of motion for the cork and, by substitution or otherwise, show the motion of the cork is $x=c\sinh (bt)$ and give expressions for $c$ and $b$ in terms of $P_g,?,l$ and $v_0$. Show that, if $v_0$ is negligibly small, the cork exits the bottle with velocity $v=\sqrt{\frac{{\mathbf{P}}_g}{?}}.$ If the density of cork is $0.5\mathrm{g}{\mathrm{cm}}^{?3}$, and a typical champagne bottle is pressurized to $400\mathrm{kPa}$ above atmospheric pressure, find (neglecting air resistance) to what height it is possible to fire a cork. An actual champagne cork does indeed start with a length $l$ within the bottle, but also has a stopper on the end that doubles the mass of the cork. How will this modification change the maximum achievable height?