# interesting problem in fluid mechanics

• Aug 22nd 2006, 02:07 PM
bobbyk
Acceleration of sphere
Here's a rather interesting problem in fluid mechanics.
A hollow weightless infinitely thin spherical shell (yes, this IS an ideal problem)
is evacuated and tied with a string to the earth's surface, where the gravitational acceleration is g. Of course, there will be an upward tension in the string of m*g, where m is the mass of the displaced air.
The string is then cut. Assuming the air is an ideal gas, what is the INITIAL
upward acceleration of the sphere? Some would say g, but this is wrong,
since it doesn't account for the acceleration of the air adjacent to the
sphere.
The problem was reportedly given to Feynman, and, after working on it for a
number of hours, came up with the answer. I won't tell you now what he got, but will leave it for those who might want to try to solve it. I don't know how Feynman did it, but will post his answer later on.
• Aug 22nd 2006, 08:36 PM
CaptainBlack
Quote:

Originally Posted by bobbyk
Here's a rather interesting problem in fluid mechanics.
A hollow weightless infinitely thin spherical shell (yes, this IS an ideal problem)
is evacuated and tied with a string to the earth's surface, where the gravitational acceleration is g. Of course, there will be an upward tension in the string of m*g, where m is the mass of the displaced air.
The string is then cut. Assuming the air is an ideal gas, what is the INITIAL
upward acceleration of the sphere? Some would say g, but this is wrong,
since it doesn't account for the acceleration of the air adjacent to the
sphere.

I won't answer the main part of this question as I know what the added mass
for a sphere is under these conditions. But I will discuss the expectation that
the acceleration would be -g.

The simple minded analysis would be: In the absence of added mass the
force acting on sphere is -mg, where m is the mass of displaced air,
and there is a minus sign because g is downwards acting and the
buoyancy force is upwards. The mass being accelerated is near zero, so
the acceleration should be very large, and not -g. (identically zero
mass as required by the thought experiment would give infinite acceleration).

Quote:

The problem was reportedly given to Feynman, and, after working on it for a
number of hours, came up with the answer. I won't tell you now what he got, but will leave it for those who might want to try to solve it. I don't know how Feynman did it, but will post his answer later on.
RonL
• Nov 18th 2006, 02:40 PM
bobbyk
Sphere acceleration
You are the only one to have answered my question about the sphere
acceleration, so I will ask you in a very friendly way, will you post your
answer, and then I'll tell you what Feynman found.
Look, this is not a competition --- I think you are one of the greatest
on this forum!
• Dec 6th 2006, 06:43 PM
bobbyk
Sphere acceleration
OK, since you don't want to post your solution, I'll tell you what Feynman
found. The initial upward acceleration is 2g.
My friend's brother was at Caltech and a student of Feynman and he told
• Dec 6th 2006, 08:06 PM
CaptainBlack
Quote:

Originally Posted by bobbyk
OK, since you don't want to post your solution, I'll tell you what Feynman
found. The initial upward acceleration is 2g.
My friend's brother was at Caltech and a student of Feynman and he told

Sorry I didn't reply before, I missed the previous post (presumably it occured
either at a time when I was preoccupied with other things or partiularly
p***** off with MHF).

The added mass of a sphere is:

$
M_a=\frac{2}{3}\pi\, r^3\, \rho
$

which is half the mass of the displaced air, so the upward accelleration is twice that
expected of a body of mass equal to the mass of displaced air subject to the same
boyancy force, or $2g$

RonL

(however I would like to know how Feynman deduced this, as the added mass
calculation is not something I can do in my head.)
• Dec 6th 2006, 09:27 PM
bobbyk
Sphere acceleration
Great, Captain, it increases my admiration for you, and would increase it
even more if you had replyed before I told you what Feynman found!
(only kidding)
• Dec 6th 2006, 10:00 PM
CaptainBlack
Quote:

Originally Posted by bobbyk
Great, Captain, it increases my admiration for you, and would increase it
even more if you had replyed before I told you what Feynman found!
(only kidding)

If Feynman's solution did not depend on the form of the cavity
being a sphere, he will of course have had a fallacious argument.

In the case of a cylinder the added mass is the mass of the displaced
air/fluid. in which case the upward acceleration would have been g

RonL
• Dec 6th 2006, 10:22 PM
bobbyk
Sphere acceleration
Sadly, since my friend, his brother, and Feynman are all dead now, I have no way of finding out how Feynman did it. He was one of the greatest, though,
as I'm sure you'll agree.
• Dec 7th 2006, 12:01 AM
CaptainBlack
Quote:

Originally Posted by bobbyk
Sadly, since my friend, his brother, and Feynman are all dead now, I have no way of finding out how Feynman did it. He was one of the greatest, though,
as I'm sure you'll agree.

Which is why I am interested in how he tackled this. His methods are always illuminating

RonL