Relating Sink Time to Hole Diametre- Water Bowls

[LemoneyF]

Homework Statement

Hi Guys

This is school physics problem, but it doesn't really fit the criteria of a homework post as its not a set question and I am not after an answer, just an idea of where to go next. The task was completely up to us, so I had to go and unknowingly pick the one that would brutally mindf**k me. I know its something so simple, something obvious I am not seeing. I could be way off, who knows.

The aim of the investigation is to relate the sink time of saxon bowls in water, (supposedly used back in their time to give a person a set time to speak in their meetings and what not) to the diametre of the holes drilled in the bottom of them. From what I understand the theory behind it will largely be based upon Bernoulli's Equation,
P1 + 1/2pv1^2 + pgh1 = P2 + 1/2pv2^2 + pgh2

So taking equally sized PVC pipe end caps with differing sized holes drilled in their centres; times taken for them to competely submerge beneath the surface of water were taken. Temperature and density were kept constant.

Holes ranging from 6-16mm were drilled, with all 'bowls' having volumes of 61ml. The sink times followed a roughly exponential pattern when averages were taken and they were graphed.

Homework Equations

P1 + 1/2pv1^2 + pgh1 = P2 + 1/2pv2^2 + pgh2
Flow rate Q= A/V

The Attempt at a Solution

So I've modeled this so far by looking at the velocity of the water entering the hole (and subsequently the fill rate), caused by the pressures of water acting on the bowl (buoyant force); and the velocity of the bowl moving downwards in relation to the water, caused by the combined forces of air pressure and weight. I have half a notebook full of scribbled notes but haven't really gotten much further than the basic concept. And I know this is post is probably missing information still. Can anyone help? :s

 

[pcm]

apply bernoulli theorem at points i) just inside open end and ii) just inside hole.to the first approximation take the ratio of area of hole to that of open surface to be negligible, to find the velocity of outcoming water.now use continuity equation to find the velocity of water at the open surface.relate this velocity to the rate of fall of water level in the bowl...doing some integration you can find the time to get empty.

for small holes time is inversely proportional to square of diameter.

 

[LemoneyF]

apply bernoulli theorem at points i) just inside open end and ii) just inside hole.

Ok, so using Bernoulli theorem I've made P1 air pressure, representing the only (?) force acting to oppose the water filling the bowl. I've made P2 the buoyant force. Given the height below the waterline would never be constant, could i not use that element of the equation?

now use continuity equation to find the velocity of water at the open surface.relate this velocity to the rate of fall of water level in the bowl.

I watched a lecture online about this and it was quite helpful, thanks. Although, I can't get any reasonable answers out of it :c

Taking some of my results, I tried to get the velocity of water rise inside the cup but got something way too big, and I am not sure what I am doing wrong.

Using A1V1=A2V2 , with one half representing conditions in the hole and the other in the cup.

Internal Cup diametre- 60mm (A= 2.827e-3)
Hole diametre- 6mm (A= 2.827e-5)
Time taken to Sink- 11.5s
Volume- 61ml

5.3ml/s - 0.0053L/sec

Q=Av, gives water velocity in the 6ml hole of 187.5m/s (huge). Going on though, this gives cup water rise velocity of 1.87m/s after rearranging to original equation to A1V1/A2.

Does anyone have any ideas?