METR 3113 Practice Problems for the Final Exam 2006
Version from 6:24 pm December 6. Correction to answer in 3 b. Here are some practice
problems for the post-Quiz-3 material: more about pressure and buoyancy, Archimedes’ prin-
ciple, continuity equation, and Bernoulli’s equation. A portion of the Final Exam (Thursday,
December 14, 8 am) will be devoted to this material and to the recent JiTTs.
1. (a) Water ﬂows from a circular hose that has a radius r. The water exits the hose
with uniform speed v. The hose is used to ﬁll a circular pool of radius R to a
depth of h. How much time is required to ﬁll the pool, if the pool was initially
dV /dt = πr2 v = dz/dt πR2 , t = R2 h/r2 v
(b) The water in the hose is coming from an unpressurized tank with the water surface
at a height L above the end of the hose that is ﬁlling the pool. (The end of the
hose is held above the water in the pool). Assuming ideal ﬂow with no viscosity,
how √ the speed v of the water, coming out of the end of the hose, related to L?
v = 2gL
(c) A rubber ducky is ﬂoating in the pool of water with density ρw . The rubber ducky
has a uniform density of ρd , less than that of water. What fraction of the volume
of the ducky is below the water line? ρd Vd g = ρw Vb g, ρd /ρw = Vb /Vd
2. (Final Exam 2005) The density of ice is slightly less than the density of water. That is
why ice ﬂoats. Consider an ice cube ﬂoating in a glass of water. Prove that when the
ice melts the level of the water in the glass does not change. Use Archimedes’ principle
in the proof. (The proof is relevant to meteorology; it is why we know that the melting
of ice ﬂoating in the Arctic Ocean will not raise the sea level.)
Archimedes: Mi = ρw Vb . But also Mi = ρw Vmelted , so Vmelted = Vb .
3. (a) A rock is suspended by a string. The weight of the rock is 10 N. The density of
the rock is 4 times that of water. The rock is dipped into water. What is the
magnitude of the tension FT in the string, while the rock is suspended in water?
Give the answer in units of Newtons.
FT =7.5 N
(b) A rock is suspended by a string. The weight of the rock is 10 N. The density of
the rock is 4 times that of water. The rock is dipped into a ﬂuid, and the attached
spring scale measures a weight of 8 N. What is the ratio of the density of the ﬂuid
to the density of water? ρr /ρf = W/B = 5, but ρr = 4ρw , so ρf /ρw = 5 .
4. (a) An object in the shape of a rectangular parallelepiped (a fancy phrase for “box”)
of volume V is submerged in a hydrostatic ﬂuid of constant density ρF . For the
situation where two faces are perpendicular to gravity, derive the fact that the
upward force of buoyancy is FB = gρF V . A sketch could be useful. (Note: This
FB also applies to objects of general shape).
(b) Suppose the “object” studied above is a parcel of air with density ρp . Show that
the net vertical force, the sum of the weight and buoyancy, is proportional to the
diﬀerence in density.
(c) Suppose the ﬂuid is air at temperature TF and the “object” is a parcel of air at
temperature Tp . Show that the sum of the weight and buoyancy is proportional
to the diﬀerence in temperature. (Note that in meteorology this net vertical force
is sometimes referred to as simply “buoyancy”).
(d) Using the ideal gas law, show that the net vertical force per unit mass, meaning
the vertical acceleration, is equal to:
Tp − TF
5. (a) Derive Bernoulli’s equation for the axis of a horizontal pipe, and a constant density
ﬂuid. Speciﬁcally, beginning with
du 1 dp
=− , (2)
dt ρ dx
+ = constant . (3)
(b) Suppose we dare to interpret u in (3) as the magnitude of the velocity vector, and
not just the velocity component in the ı direction. (In fact, such an interpretation
is valid, but we have not derived it). Suppose we apply (3) to a suction vortex
of a tornado, with air being sucked into the low pressure of the core of a tornado
at the base of the tornado. Suppose the core pressure is 50 mb less than the
environment. What is speed of the parcel that has been sucked into the vertical
jet in the core of the vortex?
u = 100 m s−1