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 Problem Statement
 A cylinder, open on one end and sealed on the other except for a small hole of varying radii. It will be placed in a bucket of water from the side of the hole, and the time taken for the cylinder to completely submerged will be measured.
 Relevant Equations

##F_{cylinder}=F_{gravity}F_{buoyant}F_{drag}##
##F_{ext}+v_{rel}\frac{\mathrm{d} m_{total}}{\mathrm{d} x}=m_{total}v##
I'm trying to write up some theory for this experiment I'm carrying out. I'll be changing the radius of the hole at the bottom of the cylinder, and I think the time taken for the cylinder to completely submerge is inversely proportional to the size of the hole. Problem is that I'm stuck at deriving this, so any help is much appreciated!
First I have equated all forces acting on the cylinder, and simplified it:
##F_{cylinder}=F_{gravity}F_{buoyant}F_{drag}##
##F_{cylinder}=m_{cylinder}g\frac{1}{2}C\rho v^2A##
I found the force of the water onto the cylinder and then equated to the net force:
##F_{water}=\frac{\mathrm{d} p_{water}}{\mathrm{d} x}##
##F_{net}=m_{cylinder}g  \frac{1}{2}C\rho v^2A + \frac{\mathrm{d} p_{water}}{\mathrm{d} x}##
Then I derived and used the variablemass equation:
##F_{ext}+v_{rel}\frac{\mathrm{d} m_{total}}{\mathrm{d} x}=m_{total}v##
I equated F_{ext} to F_{net}:
##m_{cylinder}g  \frac{1}{2}C\rho v^2A + \frac{\mathrm{d} p_{water}}{\mathrm{d} x}=m_{total}vv_{rel}\frac{\mathrm{d} m_{total}}{\mathrm{d} x}##
And then I'm stuck and I have no idea what to do next...if there is anything missing please tell me. The issue is that the mass of water, as well as the downwards velocity of the cylinder, is changing at a rate I do not know.
First I have equated all forces acting on the cylinder, and simplified it:
##F_{cylinder}=F_{gravity}F_{buoyant}F_{drag}##
##F_{cylinder}=m_{cylinder}g\frac{1}{2}C\rho v^2A##
I found the force of the water onto the cylinder and then equated to the net force:
##F_{water}=\frac{\mathrm{d} p_{water}}{\mathrm{d} x}##
##F_{net}=m_{cylinder}g  \frac{1}{2}C\rho v^2A + \frac{\mathrm{d} p_{water}}{\mathrm{d} x}##
Then I derived and used the variablemass equation:
##F_{ext}+v_{rel}\frac{\mathrm{d} m_{total}}{\mathrm{d} x}=m_{total}v##
I equated F_{ext} to F_{net}:
##m_{cylinder}g  \frac{1}{2}C\rho v^2A + \frac{\mathrm{d} p_{water}}{\mathrm{d} x}=m_{total}vv_{rel}\frac{\mathrm{d} m_{total}}{\mathrm{d} x}##
And then I'm stuck and I have no idea what to do next...if there is anything missing please tell me. The issue is that the mass of water, as well as the downwards velocity of the cylinder, is changing at a rate I do not know.