Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: [Phys-L] Another Fluid/Density 'Problem'



This is a very cool problem Chuck. Thanks for posting it. Assuming the volume of the milk+cream doesn't change, the pressure at the bottom of the container clearly has to increase--upon mixing, some of the cream that was in the central cylindrical column of milk+cream goes out of the column where it is mixed with the milk surrounding the column and it is replaced with the (denser) milk that was outside the column. Clearly the column is heavier than it was. To keep the column in equilibrium, the glass at the bottom of the column clearly has to be exerting a greater upward force on the column, meaning the pressure at the bottom of the milk bottle is greater. The question that comes to mind is not, "how could the pressure be greater," but "how could the pressure not be greater?" The problem illustrates why it is a bad idea to assume that the force that the fluid exerts on the bottom of a container is equal to the force that the bottom of the container exerts on whatever surface the container is resting on (even if the mass of the bottom of the container can be considered negligible). If the milk bottle was originally sitting on a scale, and we picked up the bottle, shook it up, and put it back on the scale, the new scale reading would be the same as the original scale reading despite the fact that the pressure at the bottom of the fluid is greater. To consider how the scale reading could be the same despite the increased pressure at the bottom, we consider a free body diagram of just the bottle. At those points in the fluid where the fluid is in contact with the sloping sides of the bottle, the pressure is greater than before because there is less cream and more milk above. The force of the fluid on the sloping sides has an upward component. The increased pressure means an increased upward force component on the sloping sides of the bottle. The increased downward force on the bottom of the bottle is exactly compensated for by the increased upward force component on the sloping sides of the bottle. We know this because the bottle is in equilibrium both before and after shaking up the milk+cream.

-----Original Message-----
From: Phys-l [mailto:phys-l-bounces@phys-l.org] On Behalf Of Chuck Britton
Sent: Wednesday, January 29, 2014 5:52 PM
To: Phys-L@Phys-L.org
Subject: [Phys-L] Another Fluid/Density 'Problem'

| |
| |
/ \
/ \
| |
| |
| |
| |
| |
| |
| |
|________|

An Old Fashioned Glass Milk Bottle with a narrow neck is delivered to your
doorstep in the morning and because it is no homogenized, the Cream has
risen up into the narrow(er) region.

The fluid pressure exerted on the bottom of the bottle is (rho) g h.
Where (rho) is the average density of the column of milk/cream that extends
from the top surface to the bottom.

Now we will thoroughly MIX the bottle of milk and we note that the average
density of this same column of milk is GREATER than (rho). Let's call this new
density (RHO).

How can the pressure have increased while the area of the bottom has
stayed constant?
How can we accept a greater amount of downward fluid force caused by
'just' mixing the milk?

(This is reprised from back in the last century.) Probably presented better
back then.
;-)
_______________________________________________
Forum for Physics Educators
Phys-l@phys-l.org
http://www.phys-l.org/mailman/listinfo/phys-l