February - Scitation

Physics Challenge for
Teachers and Students
A Pipe Dream
Challenge: An open glass tube has a length of 0.500 m.
The tube is held vertically above a water-filled container so
that the bottom end of the tube slightly touches the surface
of the water. Under these conditions, the height of the
water column in the tube is 20.0 mm.
What would be the height of the water column in the
tube if the top end of the tube is sealed before touching the
water? The atmospheric pressure is 1.01 x 10 5 Pa.
Solution: When the top end of the tube (of
inner radius r) is open, there is atmospheric
pressure at both the top and bottom surfaces
of the water column of height h = 20 mm.
Therefore its weight ρπr 2 hg (where ρ =
1000 kg/m 3 is the density of water) is balanced
by a capillary force F cap due to an adhesive
attraction between the glass and the water
around the circumference of its top surface,
F cap = ρπr 2 hg. (1)
(There is also an adhesive attraction between
the curved side of the water column and the
glass, but this produces no net force on the
water.)
If the top end of the tube is sealed before its
bottom end is immersed, we trap an air column
of initial length L = 500 mm that we can
model as an isothermal ideal gas. Suppose this
time the water column rises to height h' inside
the tube. This compresses the air column to a
gauge pressure of P, where
PV i i = PfVf ⇒ PatmLπr = ( P + P)( L− h′ ) πr
atm
⇒ P= P
Solutions to February 2006 Challenge
2
2
h′
P
L− h′
≈
atm atm
h′
,
L
(2)
with P atm = 101 kPa atmospheric pressure.
Now we have four forces acting on the water
column: the difference between atmospheric
pressure upward on its bottom and the air column
pressure downward on its top multiplied
by the inner cross-sectional area of the tube,
the same capillary force upward on its top circumference
as before, and its weight downward.
The force balance thus becomes
ρπr 2 h' g + Pπr 2 = F cap . (3)
Substituting Eqs. (1) and (2) and rearranging
gives
⎡ ⎤
′ = ⎢ + ⎥ (4)
⎣ ⎦
=
−1
Patm
h h 1 0. 925 mm.
ρgL
Note that h'

lege, Cleveland, OH)
Art Hovey (Milford, CT)
José Costa Leme (High School Lanheses, Viana
do Castelo, Portugal)
Stephen McAndrew (Trinity Grammar School,
Summer Hill, NSW, Australia)
Leo H. van den Raadt (Heemstede, The Netherlands)
Many thanks to all contributors and we hope to
hear from you in the future!
Please send correspondence to:
Boris Korsunsky
444 Wellesley St.
Weston, MA 02493-2631
korsunbo@post.harvard.edu
THE PHYSICS TEACHER ◆ Vol. 44, May 2006 29