# Measuring the spring constant in a system with hydrostatic buoyancy

I have a physics lab tomorrow and I would appreciate it if you could give me your opinion on whether my reasoning is correct for this situation or not:

https://i.sstatic.net/WMzA7.jpg

The title of the lab is hydrostatic and is roughly described in the attached pic. We will take data using different weights to measure the deformation of the spring and the increase in the level of the water (yes, the fluid is water) and the thing that we have to find is the spring constant "k" but (and my professor was very clear about it) for the whole system.

Now I am having some doubts, but I think that he means that we have to find some function of F that uses d (deformation of the spring) and h (increase in water height) and then we'll use least squares (done this in previous labs) to find the constant k, what I'm currently thinking is

F = restitution force of the spring

m = mass of the submerged body
d = deformation of the spring h = increase in water level
W= weight of the submerged body
B = buoyant force
V= volume of the displaced liquid / small body

F=−k*d
B=d(water density)Vg

W=m*g

Taking upwards as positive

F+B−W=0

but now I have to find either a) the volume of the displaced fluid or the volume of the body, both of these are not data (or can't be found without the dimensions of the container or the small body as far as I know) at all, the Buoyant force has to be smaller than the weight since it's at equilibrium only after the spring force begins acting

Is my current train of thought correct?

1) since the difference in water level changes with different masses I think it's safe to assume that the body to be submerged is not completely inside from the beginning, since if it were that way after adding more masses the total displaced volume would not increase

II) Still having troubles with the F = W - B part, since I really can't figure how to calculate the Buoyant force without the volume of the object or the water displaced and I can't find the one of the water displaced since I do not know the dimensions of the container.

III) the most smart course of action seems to be to find F using only B and W and then use that F with d (spring deformation) to find ''k"

IV) I can't grasp what would happen if the mass were to get so big that the body would be completely submerged and "h" would stop growing but it's probably not going to happen in the laboratory

I have a feeling I'm missing something really obvious and even a small push in the right direction would be nice

• This is actually a really cool question because it is used as an example in renormalization theory. That is, see equation (1) through (2) in: arxiv.org/abs/hep-th/9912092 Commented Feb 26, 2011 at 0:51

This probably isn't going to be in time for your lab tomorrow, but I'll attempt to shed some light on this:

In this sort of fluid-structure interaction with an oscillating/vibrating body, there are generally three fluid effects that need to be considered: added mass, added stiffness and added damping.

Added mass is basically the fluid's mass adding to that of the oscillator, because fluid has to be accelerated to move it out of the way of the moving object.

Added stiffness is fluid effects adding to the stiffness of the oscillator (I'll expand on that shortly).

Added damping is extra damping of the oscillator, caused by viscous friction of the object as it moves through the fluid.

The first two will tend to change the resonant frequency of a mechanical oscillator, as this is usually given by $\sqrt{\frac{k}{m}}$. Added damping will obviously increase the damping of the system and cause oscillations to subside more quickly, as it helps to dissipate energy.

So, in terms of the effect of the fluid on the spring constant 'k' in your experiment, it's the added stiffness that you should be looking at. So, what behaviour of a fluid will provide an opposing force that's proportional to the amount of displacement? The answer is buoyancy with a free surface. If an object is partially submerged, then the buoyancy force will increase as it submerges further and reduce as it lifts back up out of the fluid. If the object is completely submerged, then the buoyancy won't change with vertical displacement, so then there shouldn't be any added stiffness.

So, my prediction is that k will be higher if the object is partially submerged and will be unchanged if the object is completely submerged.