Rigid body dynamics. As a SpaceX engineer ${?}^1$ you must calculate the pressure in the fuel tank of

a Falcon $9$ rocket in various launch stages $($prior to launch, during launch, and post$-$launch$).$ While

in reality the fuel and oxidizer are held separately, for this problem we'll imagine the fuel tank as a

single large cylinder full of liquid. Initially, the engine is at rest and stands upright on the launch

pad as shown. The diameter of the fuel tank is $D=2\mathrm{.}5m$ and its height is $H=30m.$ The fuel

has density $=900kg*m^{-3}$ and the initial depth of the fuel is $h_1=25m.$

a$.$ What is the $($gage$)$ pressure, in kilopascals, at the bottom of the fuel tank prior to

launch? $($Assume the top of the tank is vented to the atmosphere.

b$.$ After launch, the peak acceleration is $a_z=25m*s^{-2}($at this point, the fuel depth is

$h_2=4m).$ If the rocket stays vertical $($and $a_x=a_y=0),$ what is the gage pressure

in kilopascals at the bottom of the tank during peak acceleration?

c$.$ Dragon $9$ boosters are reusable, which is a gamechanger for commercial space flight.

They generally land upright on a "drone ship". However, just before landing one of

the stabilizing thrusters malfunctions and the rocket starts spinning steadily around its

vertical axis just before landing. There is not much fuel left $(h_3=0\mathrm{.}5(m)).$ If the floor

of the fuel tank is exposed, some sensitive equipment will be damaged or destroyed.

What is the maximum angular velocity $,$ in Hz $,$ where the tank floor is still covered

with fuel? At this point $a_z$ is negligible, and no fuel is being burned. Hint: consider the

free surface of the fluid, which is experiencing centripetal acceleration and will therefore form a

paraboloid. Mass is conserved from the un$-$spun$-$up state where the surface is flat.$)$

$($a$)$

$($b$)$

$($c$)$