TUTORIAL SOLUTION SHEET 3 – Pressure
A. 1.
Concepts:
- Pressure forces act
inwards from all
directions
- Pressure and depth
Relevant Equations: = ! +
P: Pressure !: Pressure at surface : Density of fluid d: depth " =
Diagrams:

Solution:
The pressure you experience when submerged in a fluid
(including the atmosphere, remember tute 1!) causes forces
that press inwards on you from all directions. Furthermore,
pressures add. This way, deep down in the ocean the
pressure is much greater than that of sea level. This is
because the atmospheric pressure, !, is increased by the
weight of the water for a distance above you (this is the part of the pressure equation, see Figure 1). Human
lungs are mainly designed to withstand the pressure
differences caused by your diaphragm, allowing air to be
drawn in and pushed out from them. When your diaphragm
pulls down to expand your lungs (Figure 2), you now have
a larger area for the same amount of air, and so the
pressure, #, goes down inside your lungs. This draws in
air from the pressure difference (# < !) created between
your lungs and the air outside, and vice versa for breathing
out.

As a diver goes further down in the ocean, the added
pressure from the water above causes a large pressure
difference between the inside of their lungs and its
surrounds. To be able to breath at all, divers are
accompanied with an oxygen tank and pressure regulator
that keeps the pressure in their lungs at a safe level, $. If
they go too deep, however, their lungs will eventually
collapse under the added pressure from the ocean! It is
usually recommended to try to avoid this effect.

The pressure forces also compress the air in your blood to
much smaller volumes. As a diver rises to the surface of the
ocean, the air in your blood begins to expand as the
pressure decreases. If you rise too fast, and don’t allow
enough time for this extra volume of air to escape your
blood stream through your lungs, bubbles form in your
blood. This can be extremely painful and even cause death
if the air bubbles reach your heart. When diving, one must
surface slowly to avoid getting the bends.


Figure 1: Breathing in
Figure 2: Diving pressures
Figure 3: Pressure Regulator
B.
Concept:
- Pressure at
points of equal
height in a fluid
at rest are equal

Relevant Equation: = ! +
Solutions i):
Water is less dense than mercury and cannot mix with it. We
learned in last week’s tutorial that the buoyant force caused by a
volume of liquid increases with the liquids density (remember that
we compared air to water with the submerged scales question). In
this way, the mercury exerts more upward force on the water than
the water would on the mercury, and so the water floats and the
mercury stays at the bottom.
Diagrams ii): Solutions ii):
The key to this problem is that the pressure at every point at a
given horizontal level in a single body of fluid at rest must be the
same. The mercury in this question is a single body of fluid, so the
pressure at the height of the lowest mercury surface is equal to the
pressure at the same height in the other arm of the U-tube. In each
of the arms, the pressure is caused by the above fluid (including
the air, which is at atmospheric pressure, !, and so is equal for
both sides of the tube): %&'( = )*+,( [ℎ]-+ + ! = [ℎ]-!! + ! -+ℎ = -!!ℎ-!.
ℎ = -!.ℎ-!.-+ = 1.00 × 10/ kg.m0/ × 0.050 m13.55 × 10/ kg.m0/ ℎ = 0.0037 m = 0.37 cm
Diagrams iii):

Solutions iii):
We use the exact same method as above in ii) to find 123425: %&'( = )*+,( [ℎ]%*67*8 + ! = [ℎ]-+ + [ℎ]-!. + ! [ℎ]%*67*8 = [ℎ]-+ + [ℎ]-!! %*67*8 = -+ℎ + -!.ℎ-!.ℎ%*67*8 = 13.55 × 10/ kg.m0/ × 9.4 × 100/ m + 1.00 × 10/ kg.m0/ × 5.0 × 1009 m3.0 × 1009 m
= 5.9 × 10/ kg . (2 sig figs)
Comparing with the two possible elements, the liquid metal is
gallium.

There is an easier way to determine the unknown metal between
gallium and caesium as well. Caesium is an extremely volatile
element that would explode when put in contact with the water and
even spontaneously combusts in air! Unfortunately, however,
explosions are no longer an accepted method of diagnosis in the
University of Sydney.