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Fluid mechanics
Pressure at depth
In a cylindrical column of
water, as in any cylinder
the volume is the height
x cross sectional area
The density of
the water =
mass ÷volume
Pressure at depth
The mass of water =
density x volume
=
ρ
density x height x area
The weight of water =
mass x gravity
= density x height x area x gravity
ρhAg
(ρ = density)
Pressure at depth
Pressure = force ÷ area
(weight ÷ area)
=(density x height x area x
gravity) ÷ area
= Density x height x gravity
(area cancels out)
Pressure = pgh
Pressure at depth
p
Pressure = pgh
Since the increases
uniformly with depth
The average pressure
h
= pgh/2
or pgh
A
pgh
(h = mean depth)
Pressure = force ÷ area
Pressure at depth
Force is pressure x area
F = pgAh
p
This is the hydrostatic thrust
acting on the side of the
cylinder
h
F
F
A
pgh
The point at which it acts
Is 2/3 the depth from
the surface of the water
(1/3 from the bottom)
Archimedes
F1
If a mass is suspended By a cable
in air the tension force in the cable
is equal to the weight of the mass
F1 = mg = (ρVg)body
m
mg
where ρ is the density of the body
V is the volume of the body
g = gravity
Archimedes
If a body is totally or partially
submersed in water (or other liquid)
it will displace some of the water (the
water level will rise). The volume of
water displaced is the same as the
volume of the water
F1
m
Fmg
Fup
Archimedes
The body will experience an
apparent loss in weight which is
equal to the weight of water (or other
liquid displaced) This apparent loss
in weight is equal to the up-thrust
force of the liquid on the body
F2
m
Fmg
Fup
Archimedes
F2 = Fmg – Fup
F2 = (ρVg)body – (ρVg)water
where ρ is the density of the body
V is the volume of the body
g = gravity
and
ρ is the density of the water (liquid)
V is the volume of the water(liquid)
F2
m
Fmg
Fup
Flow through a tapered pipe
Volume flow = Volume/time (m3/s)
Flow
Mass flow = Mass /time (kg/s)
Volumetric flow symbol V
V = Av (m3/s)
m = ρ V Av (kg/s)
A = area
v= velocity
Ρ = density
Flow through a tapered pipe
Volume flow = Volume/time (m3/s)
Flow
Mass flow = Mass /time (kg/s)
Volumetric flow symbol V
V = Av (m3/s)
m = ρ V Av (kg/s)
A = area
v= velocity
Ρ = density
Flow through a tapered pipe
Rearranging for v
Flow
v=
A1
A2
V
ρA
Since V is the same at inlet area A1 and area
A2 and ρ is the same we can calculate the
velocity of flow at each by using the two
different areas in the equation
A = area
v= velocity
Ρ = density
Flow through a tapered pipe
Rearranging for v
Flow
v=
v1 =
A
A2
A1
V
A1
V
v2 =
V
A2
A = area
v= velocity
Ρ = density
Area = πr2