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The concepts of atmospheric pressure and buoyancy in gases, focusing on the differences with liquids. The density variations in gases, the concept of an 'empty' container filled with air, atmospheric pressure and its relation to height, and buoyancy forces. It also includes interactive clicker questions to test understanding.
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What determines the thickness of our atmosphere? Balance between:kinetic energy of molecules
vs
gravity
spreads molecules apart
holds molecules near earth
Consider extremes:(i) If very little gravity (eg on moon), then molecules would move, collide, and
eventually disappear into space. So no atmosphere. (ii) If gravity very strong
c.f.
kinetic energy (eg on a remote planet), molecules
move too slowly, and form a liquid or solid, like the planet itself – so againno atmosphere. Earth – balance between the two effects, so we do fortunately have an
atmosphere! (we can breathe!!)
-^ Exactly how tall is the atmosphere?
Not a meaningful question, since it gets thinner and thinner asyou go higher and higher. Even in interplanetary space, haveabout 1 gas molecule (mostly hydrogen) every cubic meter.
Air is least dense uphere… and most dense here
Unlike water, density of atmosphere varies with height, so pressurerelation in terms of depth is not as simple. Not uniform.
-^
At sea level, 1 m
3 of air has mass of 1.25 kg
At 10km height, 1 m
3 of air has mass of 0.4 kg
(this is why need additional mass of air to pressurize airplanes).^ Recall Pressure = Force/area = weight/area.So to find pressure at sea level, need tocalculate weight of a column of air rising up to“top” of atmosphere, say about 30 km.Find that a 1m
2 area cylinder, 30 km high, has
mass of 10 000kg.i.e. weight of 100 000 N.So pressure = 100 000 N/ (1 m)
2
= 100 kPa
Precisely, sea-level atmospheric pressure = 101.3 kPa
1 Pa = 1 N/m = Pascal
Measure pressure of atmosphere
-^
Simple mercury barometer: Fill tube with mercury and then turnupside down into dish. Mercury runsout into the dish until level in tube is 76cm, as shown.
vacuum
Why 76cm?Because, of pressure balance: barometer balances when weight of liquidin tube exerts same pressure as atmosphere outside.It’s 76cm, regardless of how wide the tube is:
weight of any 76cm
column of mercury equals weight of same width column of 30 km of air. If atmospheric pressure increases, then air pushes down harder on themercury , so column pushed up higher than 76 cm.
-^ How about a barometer made of water?Why not – but how tall would the glass tube have to be?
The weight of the water column would need to be the sameweight as 76cm column of mercury, but density of water is 13.6 xless than the density of mercury – hence, water barometer wouldhave to be (at least) 13.6 x 76cm = 10.3 m tall. Again, regardlessof tube’s width.
-^
Just like barometer, when you drink through a straw, it’s the
An object surrounded by air is buoyed up by a force equal to theweight of the air displaced
.
c.f.
Archimedes principle for liquids in the previous chapter.
-^ An object will rise in air (ie float upward) if its density is less than air’s density:Why? (c.f. sinking vs floating in previous chapter)Downward grav force (= weight-density x volume) is then less than upwardbuoyant force (= weight-density-of-air x volume). So there is a net upward force.Eg. He-gas filled balloon (or heated air balloon – since hot air is less densethan normal air)
Greater buoyancy if the helium could beevacuated – but not practical since how wouldkeep the balloon sides from collapsing in?Could use stronger material but then weight istoo large, so wouldn’t rise at all…
-^
Important differences: (i)
due to the air density becoming less as you go higher (liquid densityremains about the same). So
buoyant force
decreases as you rise
in
atmosphere (but stays same while rise in water). (ii) there is no “top” to the atmosphere (it just keeps thinning out), unlike liquid
surface.
Consider an air-filled balloon weighted so thatit is on the verge of sinking—that is, its overalldensity just equals that of water.Now if you push it beneath the surface, it will^ 1. sink.2. return to the surface.3. stay at the depth to
which it is pushed.
IMPORTANT NOTE: the balloon is compressible.
When you increase the pressure of a confined gas, how does thevolume change? And vice-versa? This is Boyle’s law: i.e. - If you halve the volume of container, the pressure is doubled,since more collisions (bouncing) between molecules and with walls.
P^1
V^1
= P
V 2
2
for a fixed temperature.
Effectively, the density is doubled.pressure ~ density (at fixed temp)
proportional to
Notes: (i) fixed temperature means fixed average speed of molecules(ii) strictly speaking, Boyle’s law applies to “ideal gases” – i.e. when neglect any“sticky” forces between molecules and treat them as point particles.At normal temps and pressures, air is well-approximated to be an ideal gas.
So far, talked about stationary fluids (
hydrostatics
). When fluids are
moving, (
hydrodynamics
), have additional effects.
Consider water moving through pipe of varying thickness: The volume passing through any cross-sectionis the same in a given time interval.So, in narrower region, speed must be faster.Eg. Squeeze on end of garden hose, waterspeeds up.Eg. River entering a narrow gorge speeds up.
Water flows faster here
-^ Streamlines –(eg thin lines above) represent paths (trajectories) ofparts of fluid. So are closer together in narrower regions where flow isfaster.