PHY 53
Summer 2010
1.
The king ordered a crown of solid pure gold, but what he got didn’t feel right, so
he asked Archimedes to verify that it was really solid and all gold. Archimedes
realized he needed a way to measure the density of the crown, but he couldn’t
damage it. There are various stories about what he did. The most likely is this.
He balanced the crown with an equal weight of pure
gold as shown. Then he lowered the whole system
slowly into water and watched to see if the items
were still balanced.
How did this allow him to make a determination of
whether the king had been cheated? [He had been.]
2.
crown
gold
Questions about hydrostatics.
a.
The teapot effect. It is well known that if you pour a liquid too
slowly from a spout it will double back under the spout and
a
may miss the container you are pouring into. In the drawing
fluid comes slowly out of the opening and runs back
underneath the pipe. Adhesion makes it turn the corner, but what holds it
against the bottom of the pipe for such a distance? [Consider the pressure
at the top and bottom of the stream at point A.]
b.
Four u-tubes are partially filled
with two different fluids as
shown. The tops of the columns of
fluid are open to the air. One case
shown is impossible. Identify it,
and for the others state whether
the red fluid is more dense, less
dense, or equally dense. Give your
reasons.
c.
Assignment 12
a
b
c
d
The giraffe’s head is 2 m above its heart, which is 2 m above the ground.
Discuss and explain two things about this animal: (a) The skin around its
lower legs is very tight and thick; (b) When it drinks from a source at
ground level it slowly spreads it legs before lowering its head.
1
PHY 53
3.
Summer 2010
The Magdeburg hemispheres. In the 1650’s the mayor of the German town of
Magdeburg, an amateur scientist, invented the first vacuum pump. To
demonstrate the power of the vacuum he could create, he pumped air out of a
device consisting of two metal hemispheres with a good seal between them and a
valve to pump out the air. When he had reduced the pressure inside the device as
much as he could, he closed the valve and attached each side to a team of horses
and had them try to pull the two pieces apart, which they couldn’t do.
a.
Take the difference in pressure between outside and inside as ΔP . Show
that the force necessary to pull one hemisphere away from the other is
F = π R 2 ΔP , where R is the radius of the sphere made of the two parts.
[This is an exercise in integral calculus. Consider the
force on an infinitesimal circular strip of the surface,
shown by the dark band, which has area
dA = 2π R sin θ ⋅ Rdθ . (Prove this.) Then argue that
R
θ
you need consider only the leftward component of
the inward force, which is dFx = ΔP ⋅ dA ⋅ cosθ . Do the
integral over θ ; the substitution α = sin θ helps.]
b.
Let R = 30 cm and ΔP = 0.9 atm. Find F.
[See http://upload.wikimedia.org/wikipedia/commons/5/5c/Magdeburg.jpg ]
4.
A hydraulic jack consists of a cylinder (1) with a small
cross section area a, connected by a pipe to a cylinder
1
(2) with larger cross section A. Each cylinder is fitted
with a piston that can move vertically, and the system
is filled with an incompressible fluid. A heavy load to
be raised rests on the piston in 2; the total mass of the
load and the piston is M. A force F1 is exerted downward on the piston in 1,
resulting in an upward force F2 on the piston in 2. Comment on the validity of
the following statements.
a.
The pressure in both cylinders is the same.
b.
As the piston in 1 moves down distance d the piston in 2 moves up
distance d ⋅ (a/ A) .
c.
To lift the piston in 2 and the load at constant speed, we must have
F1 = Mg ⋅ (A/ a) .
d.
The work done by F1 is equal to the work done by F2 .
2
2
PHY 53
Summer 2010
5.
A “Cartesian diver” is a small glass cylinder open at the bottom
which has a pocket of air trapped inside. It is in a sealed flexible
plastic container nearly filled with water, and initially the diver
is floating at the surface as shown. When one squeezes the sides
of the container, reducing the volume of the air above the water,
the diver sinks toward the bottom. When the squeezing force is
removed the diver rises again and floats. Explain this behavior
in detail. [What happens to the average density of the diver,
and why?]
6.
A block of mass m is made of a material with
density four times that of water. It is lowered by a
string until it is held at rest submerged in a
container of water as shown.
7.
a.
What is the tension in the string?
b.
If the container plus the water have mass M,
what is the normal force from the table on
the system?
c.
If the block is lowered to rest on the bottom, with the string slack, what is
the normal force from the table?
A tank filled with water has a small hole as shown where water emerges in a
stream. At height h above the hole there is a piston of the same cross-section area
A as the tank, resting on the water. It cannot sink because the fit is too tight for
water to flow around it, but it can slide freely down the sides of the tank as the
water level drops.
a.
If the water emerges at speed v, what is the pressure
P in the water just below the piston, in terms of v, h,
the density of water ρ, air pressure P0 and g? Ans:
d
h
P = P0 − ρ gh + 12 ρ v 2 .
b.
What must P − P0 be to support the weight of the piston, if its density is
2 ρ and its thickness is d? Ans: P − P0 = 2 ρdg .
c.
Find the speed v in terms of h, d and g. Ans: v 2 = g(2h + 4d) .
3
2. 53 An aspirator is a device that mixes small amounts of a fluid (such as
liquid 2010
PHY
Summer
fertilizer) with water, so that it can be sprayed over a large area. Shown is such a
device. Water from a hose (on the left) flows through a constriction, where a
vertical
tube
themixes
fluid
8.
An
aspirator
is introduces
a device that
(at
negligible
speed)
into
the
vv00
small amounts of a fluid (e.g., rapid
a
A
water flow. The height through
liquid
fertilizer)
water
sohitascan
which
the fluidwith
must
rise is
h
beshown.
sprayed
a large area.
As
If over
the fluid-water
mixture
exits into
thefrom
air ata speed
v0, what
shown,
water
hose passes
must
be
the
minimum
ratio
through a constriction of cross-A/a of
the cross-section
areas oftube
the two
section
a, where a vertical
regions? Assume the fluid and
introduces the fluid (moving
water have the same density !.
upward
at negligible
speed)ininto
[Find the
water pressure
the
the
rapidly
moving
water.
The
constriction.]
mixture then flows out at speed v0
through an opening of larger cross-section A. The height through which the fluid
must rise is h. Assume the fluid and water have the same density ρ , and find the
minimum ratio A/ a of the two cross-sections. [Find the water pressure in the
constriction.] Ans: (A/ a)2 = 1 + 2gh/ v02 .
9.
Questions about the Bernoulli effect in air.
a.
You can lift a card off a table by blowing horizontally over the top of the
card. Explain.
b.
In a hurricane a flat roof is removed from a building by a wind that is
blowing mostly horizontally. Explain.
c.
Race cars are designed to direct air striking the front of the car to go
underneath it. Why?
4