@bhec10 I think there are many steps to this project you’re working on. Is this a/n (thought) experiment? Start by drawing a diagram and deciding the assumptions you’re going to make (ie. is body in equilibrium etc) and the potential errors you will overlook.

There are no absolutes/certainties with such little information provided (ie so many potential calculations) , I think you need to obtain more information, and I definitely think more than a few equations are going to be used if you’re willing to make it somewhat accurate.

Anyways. Possible methods to use: (You may need to add in extra information)

By doing so, you’ll be considering forces on the cartridge, and I think mechanically, this is the best way to go on about this. Treat both bodies separately. Assume it is spherical.

Use Archimedes principle [when a body is partially or totally submerged in a fluid it experiences an upthrust equal to the weight of fluid displaced.] Also you need to know that **Volume of solid body= volume fluid displaced.**

Since viscous drag is always against direction of motion, your equation will depend on whether calculations are derived as diver sinks or swims towards the surface.

This is if diver is moving downwards:

W(solid)= U(fluid)+D (<—viscous drag)

you can keep simplifying (eg. W is a product of mass by gravity of about 9.81)

mg= mg + 6(pi)rnv (r is radius of spherical object in which case it is the cartridge, assume its perfectly spherical unless you want to immerse in more complex means of figuring this out. n is coefficient of viscosity, v is terminal velocity. pi is pi.)

Regarding stokes law, note:

“Stokes’ law makes the following assumptions for the behavior of a particle in a fluid:

Laminar Flow

Spherical particles

Homogeneous (uniform in composition) material

Smooth surfaces

Particles do not interfere with each other.”

if you want to further simplify: d is density V is volume

since m=dV and V for a sphere= 4/3pi*r^3

d*4/3*pi*r^3g(solid)= d*4/3*pi*r^3g (fluid) + 6pirnv

Substitute known values in and see what you need.

You can search things like density of water etc.

If you already have the values to begin with then just go with

W=U+ D

For the human you can resolve forces horizontally and vertically (Fsintheta)—>vert (Fcostheta)—> horiz

You can also try using Boyle’s law (gas laws)

where

P1V1= P2V2

or

P1V1/T1=P2V2/T2

p is pressure V is volume T is temperature in (K)

1 is for initial data and 2 is final conditions

but either way you’ll definitely need more information.

The project needs to be planned out thoroughly first if you want it to make any sense.

PS to find the amount of air you’ll have to work out the Volume from one of the methods. You can always assume the diver’s body and the cartridge are one big spherical object, so that you combine the diver’s information in the equation (ie the spherical object is going to weigh >85 kg, add some data to your exp) and all is hypothetically solved.

In retrospect I see that there are too many uncertainties related to this project, because the V of air won’t be constant, it’ll be instantaneously changing so you’ll need certain specialized equipment that measures V of air over frequent intervals of time. The wiki page you linked us to even has a section concerning divers:

“Divers::

Underwater divers are a common example of the problem of unstable buoyancy due to compressibility. The diver typically wears an exposure suit which relies on gas filled spaces for insulation, and may also wear a buoyancy compensator, **which is a variable volume buoyancy bag which is inflated to increase buoyancy and deflated to decrease buoyancy**. The desired condition is usually neutral buoyancy when the diver is swimming in mid-water, and this condition is unstable, so the diver is constantly making fine adjustments by control of lung volume, and has to adjust the contents of the buoyancy compensator if the depth varies.”