A simple gun can be made from a uniform cylinder of length L_{0} and inside radius r_{c}. One end of the cylinder is sealed with a moveable plunger and the other end is plugged with a cylindrical cork bullet. The bullet is held in place by friction with the walls of the cylinder. The pressure outside the cylinder is atmospheric pressure, P_{0}. The bullet will just start to slide out of the cylinder if the pressure inside the cylinder exceeds P_{c r}.
a. There are two ways to launch the bullet: either by heating the gas inside the cylinder and keeping the plunger fixed, or by suddenly pushing the plunger into the cylinder. In either case, assume that an ideal monatomic gas is inside the cylinder, and that originally the gas is at temperature T_{0}, the pressure inside the cylinder is P_{0}, and the length of the cylinder is L_{0}.
(8 pts) i. Assume that we launch the bullet by heating the gas without moving the plunger. Find the minimum temperature of the gas necessary to launch the bullet. Express your answer in terms of any or all of the variables: r_{c}, T_{0}, L_{0}, P_{0}, and P_{c r}.
(8 pts) ii. Assume, instead that we launch the bullet by pushing in the plunger, and that we do so quickly enough so that no heat is transferred into or out of the gas. Find the length of the gas column inside the cylinder when the bullet just starts to move. Express your answer in terms of any or all of the variables: r_{c}, T_{0}, L_{0}, P_{0}, and P_{c r}.
b. (9 pts) It is necessary to squeeze the bullet to get it into the cylinder in the first place. The bullet normally has a radius r_{b} that is slightly larger than the inside radius of the cylinder; r_{b}-r_{c}=\Delta r, is small compared to r_{c}. The bullet has a length h \ll L_{0}. The walls of the cylinder apply a pressure to the cork bullet. When a pressure P is applied to the bullet along a given direction, the bullet’s dimensions in that direction change by
for a constant E known as Young’s modulus. You may assume that compression along one direction does not cause expansion in any other direction. (This is true if the so-called Poisson ratio is close to zero, which is the case for cork.)
If the coefficient of static friction between the cork and the cylinder is \mu, find an expression for P_{c r}. Express your answer in terms of any or all of the variables: P_{0}, \mu, h, E, \Delta r, and r_{c}.