Height of Water in Straw vs. Time

[mruss]

Homework Statement

Problem Statement:
Imagine you had a very long straw, one end is in the ocean (sea
level) and the other end is in space. Air pressure at sea level
is a standard atmospheric pressure and space is a perfect vacuum.

Plot "height of water in straw" vs. time.

Variables:
h(t): height of water in straw (above sea level) at time t (in meters)
A: cross-sectional area of straw (in meters^2)

Starting conditions:
At time 0, height of water is 0: h(0) = 0
At time 0, water has no velocity or acceleration

Homework Equations

[/B]
F = ma
position(t) = position(0) + integral_0_t (velocity(0) + integral_0_t acceleration(t) dt) dt
Standard atmospheric pressure is ~101,000 kg / (m * s^2) per unit area
Density of water = 1,000.00 kg/m³
g = 9.8 m/s^2

The Attempt at a Solution

[/B]
Here is what I've tried and maybe someone can point out where I'm
going wrong.

net force = atmospheric pressure - force of gravity on mass of water in straw

Atmospheric Pressure
Standard atmospheric pressure is ~101,000 kg / (m * s^2) per unit area
Let's call A the cross-sectional area of the straw (in meters^2).
force of atmospheric pressure = 101,000 * A kg m/s^2

Force of gravity on mass of water in straw
Force of gravity on water depends on how much water is in the
straw (above sea level). Let's call h(t) the height of the
water in the straw at time t.

mass-of-water-in-straw(h(t)) = volume * density = h(t) * A * 1000 kg
force_of_gravity(h(t)) = mass-of-water-in-straw(h(t)) * g = 1000 * h(t) * A * g kg m /s^2

F(t) = 101,000 * A - 1000 * h(t) * A * g
m(t) = 1000 * A * h(t) (from mass of water equation above)
So acceleration (F / m) = (101 - g * h(t)) / h(t)
Written another way: a(t) = (101 - g * h(t)) / h(t)

Ok, great, so I think h as a function of time should be the solution to this equation:
h(t) = integral_0_t (integral_0_t a(t) dt) dt
Substitution my equation for a(t), I get:
h(t) = integral_0_t (integral_0_t (101 - g * h(t)) / h(t) dt) dt

Unfortunately, I don't know how to solve that... so I tried to
numerically approximate it. Let's just assume constant
acceleration over short periods of time dt, then we can use much
simpler equations and step through time in small dt steps.

Simpler equations:
a(t) = (101 - g * h(t-dt)) / h(t-dt)
v(t) = v(t-dt) + a(t) * dt
h(t) = h(t-dt) + v(t) * dt
with starting conditions h(0) = 0 and v(0) = 0 and a(0) = 0

When I try to calculate my initial acceleration
a(1), it's infinity (or rather, undefined) since the force of air
pressure is pushing on no mass (no water is above sea level
at the start of the problem)! There is obviously some flaw
in that logic, but I'm not sure what it is.

Any help is apprecated, thank you.

 

[Simon Bridge]

You appear to have neglected:
1. the mass of air initially filling the straw;
2. the rotation of the Earth
3. the falloff in the acceleration of gravity with altitude
4. distance the bottom of the straw is below the water surface - i.e. initially no water in the straw.
5. viscosity in the water ...

You may want to start with a related question - what height of water will be supported by a pressure difference of 1atmos?

Written another way: a(t) = (101 - g * h(t)) / h(t)

... it is not really useful to mix numbers and variables in an equation.

Setting up the DE - the acceleration of the com from Newtons laws - cancelling the area - you get something like:

$P-\rho gh = \rho gh \ddot{h}: h(0)=0, \dot h(0)=0$

$P$ is 1atmos, $\rho$ is density of water, $g$ accel of gravity at sea level, $h(t)$ is the distance of the top of the column above sea level ... center of mass is at h/2.

put it another way: $$\frac{d^2h}{dt^2} = \frac{P}{\rho gh}-1$$

 

[mruss]

Re your points:
1) I did neglect #1-5 as you mentioned. Do you think assuming those effects are negligible is an OK approximation?
2) I have already worked out the related question, height of water supported by 1atmos. I agree it's an interesting problem.
3) Fair point about mixing constants with variables.

I think you have a small typo in your DE. I think the correct statement is as follows:
$P-ρgh = ρh\ddot{h}: h(0) = 0, \dot{h} = 0$

Are you able to solve this (numerically or otherwise)? The problem that I am having is that if $h(0) = 0$, then $\ddot{h}=∞$.

 

[Simon Bridge]

I'm pretty sure it's a standard DE ... y(ay''+by+c)=d, here b=0.

What happens to you calculation if you start with a small initial volume of water in the straw?
Well spotted on the typo.

Anyway - the DE is 2nd order and non-linear for form y''=f(y,y')
... substitute $v=\dot h$ gives you a 1st order separable equation you can solve for v(h)
... which, in turn, gives you 1st order equation(s) for h(t).

 

[PJC01]

First of all, great job breaking down the problem and identifying the variables and starting conditions. Your approach seems to be on the right track, but there are a few things that need to be adjusted.

1. Units: Make sure all units are consistent throughout your equations. In your calculation for the force of atmospheric pressure, you have A in meters squared, but in your calculation for the mass of water in the straw, you have A in meters. This will cause issues when you try to integrate and solve for h(t).

2. Integration: When integrating, make sure to use the correct limits and variables. In your equation for h(t), the variable being integrated with respect to should be t, not h(t). So it should be h(t) = integral_0_t (integral_0_t a(t) dt) dt.

3. Initial acceleration: You are correct in identifying that there is an issue with your initial acceleration calculation. This is because your equation for acceleration assumes that there is already some water in the straw at time t. However, at time t = 0, there is no water in the straw, so the force of gravity on the water is 0. You can handle this by setting a(0) = 0 in your initial conditions.

4. Simplified equations: Your simplified equations for velocity and height are correct, but you need to make sure to update the variables with each time step. So for example, when calculating h(1), you need to use the updated value of v(1) from your calculation for v(1).

I hope this helps guide you in the right direction. Keep up the good work!