Density, Pressure – Learning Outcomes
 Define density and pressure, and give their units.
 Solve problems about density and pressure.
 Discuss pressure in liquids and gases.
 State Boyle’s Law.
 Demonstrate atmospheric pressure.
 Discuss pressure in weather and diving.
 State Archimedes’ Principle.
 Demonstrate Archimedes’ Principle.
 State the Law of Flotation.
 Demonstrate the Law of Flotation.
 Discuss hydrometers.
Density
 The density of a substance is its mass per unit volume.
 Formula: 𝜌 =
𝑚
𝑉
 ρ = density, m = mass, V = volume.
 It is a scalar quantity, measured in 𝑘𝑔/𝑚3
 e.g. A piece of wood has mass 80kg and volume 100m3.
Find its density.
 e.g. A cubic block of gold has length 120cm and density
19 300kg/m3. Find its mass.
Density
𝜌=
𝜌=
𝑚
𝑉
80
100
 𝜌 = 0.8 𝑘𝑔/𝑚3
 𝑉 = 1.23 = 1.728 𝑚3
𝑚 =𝜌×𝑉
 𝑚 = 19 300 × 1.728
 𝑚 = 33 350.4 𝑘𝑔
Pressure
 Pressure is force per unit area.
 Formula: 𝑃 =
𝐹
𝐴
 P = pressure, F = force, A = area
 It is a scalar quantity, measured in Pascals (Pa).
 e.g. Cian leans on his table with a force of 30N. If his
elbow has an area of 0.015m2, what pressure is he
exerting on the table?
 e.g. Atmospheric pressure is ~100kPa. If a circle on the
ground has radius 10cm, what force is the atmosphere
exerting on the circle?
Pressure
𝑃=
𝐹
𝐴
𝑃=
30
0.015
 𝑃 = 2𝑃𝑎
 𝐴 = 𝜋𝑟 2 = 𝜋 × 0.1
2
≈ 0.0314𝑚2
𝐹 =𝑃×𝐴
 𝐹 = 100 000 × 0.0314
 𝐹 = 3140𝑁
Pressure in a Fluid
 The pressure in a fluid increases with depth.
 Formula: 𝑃 = 𝜌𝑔ℎ
 P = pressure, ρ = density, g = acceleration due to gravity, h =
height / depth
 Pressure acts perpendicular to any surface immersed in
the fluid.
 At equal depths, the pressure is the same.
Pressure in a Fluid
 e.g. Kevy is going scuba diving off the coast of Mayo.
What pressure is the water exerting on him if he is 10m
below sea level?
 e.g. Find the pressure, due to the water, at a depth of
33m in water.
 e.g. A can of height 10 cm is
submerged in water. What
is the difference in pressure
between the top and
bottom of the can?
Pressure in a Fluid
 𝑃 = 𝜌𝑔ℎ
 𝑃 = 1000 × 9.81 × 10
 𝑃 = 98 100𝑃𝑎
 𝑃 = 𝜌𝑔ℎ
 𝑃 = 1000 × 9.8 × 33
 𝑃 = 323 400𝑃𝑎
 𝑃 = 𝜌𝑔ℎ
 𝑃 = 1000 × 9.8 × 0.1
 𝑃 = 980𝑃𝑎
Boyle’s Law
 Boyle’s Law: The volume of a fixed mass of gas is
inversely proportional to its pressure at constant
temperature.
 Formula: 𝑝 ∝
1
𝑉
or 𝑝𝑉 = 𝑘
 p = pressure, V = volume, k is a constant for a particular gas.
 e.g. A pressurised gas doubles its volume when some of
the pressure is relieved. By what factor did the pressure
change if the temperature remained constant?
 e.g. The volume of a fixed mass of gas is 600cm3 at a
pressure of 1×105Pa. Find its volume when the pressure
changes to 3.2×105Pa if the temperature remains
constant.
Boyle’s Law
 Double volume ⇒ Half pressure
 𝑃1 𝑉1 = 𝑃2 𝑉2
 1 × 105 × 600 = 3.2 × 105 𝑉2
 𝑉2 = 187.5𝑐𝑚3
Boyle’s Law
 e.g. A small bubble of gas rises from the bottom of a
lake. The volume of the bubble increases threefold when
it reaches the surface of the lake where the atmospheric
pressure is 1.01 × 105 𝑃𝑎. The temperature of the lake is
constant. Calculate the pressure at the bottom of the
lake and the depth of the lake.
 Volume increases threefold upwards ⇒ pressure is three
times as much at the bottom of the lake.
 ⇒ 𝑃 = 3 × 1.01 × 105 = 3.03 × 105 𝑃𝑎 (due to atm. + water)
 𝑃 = 𝜌𝑔ℎ
 2.02 × 105 = 1000 × 9.8 × ℎ (subtract 1.01 × 105 from atm.)
ℎ=
2.02×105
1000×9.8
= 20.6𝑚
Weather and Diving
 High pressure -> clear, sunny, dry, still air.
 Imagine the high pressure pushing the clouds away.
 Low pressure -> cloudy, wet, windy.
 Imagine all the clouds being pushed into low pressure areas.
 As you dive deeper, pressure increases. This causes
excess nitrogen (79% of air) to be dissolved in your
blood. If you surface too quickly, the nitrogen will form
bubbles as the pressure decreases. These bubbles are
dangerous and potentially fatal.
 The “cure” is to stay in a decompression chamber where the
pressure can be slowly decreased to normal levels.
 Divers’ air supplies sometimes have increases oxygen levels to
reduce this possibility.
Archimedes’ Principle
 Archimedes’ Principle states that a body wholly or
partially immersed in a fluid will experience an upthrust
equal to the weight of the fluid displaced.
To Demonstrate Archimedes’ Principle
1. Fill an overflow can with water until it overflows.
2. Place an empty graduated cylinder underneath the
spout of the overflow can.
3. Attach an object to a spring balance and note the
reading.
4. Immerse the object in the overflow can and note the
new reading on the spring balance.
5. Note the weight of the water in the graduated cylinder
and compare it to the difference in weight of the
object.
Result: The displaced water and the upthrust on the object
should be the same, verifying Archimedes’ Principle.
Law of Flotation
 The weight of a floating body is equal to the weight of
the fluid it displaces.
 Hydrometers are designed to float at different levels
depending on the density of the fluid it is immersed in
(the weight of the hydrometer will be displaced with less
volume in denser fluids).
 Hydrometers are used to find:
 the percentage of alcohol in beverages.
 the percentage of fat in milk.
 the density of sulfuric acid in a lead acid battery.