Chapter 13: Fluids. A fluid is a gas or a liquid. A gas expands to fill any container A liquid (at fixed pressure and temperature), has a fixed volume, but deforms to the shape of its container. The density r of any substance is its mass M per volume V :. Pressure.
A fluid isa gas or a liquid.
A gas expands to fill any container
A liquid (at fixed pressure and temperature), has a fixed volume, but deforms to the shape of its container.
The densityr of any substance is its mass M per volume V:
PressureP is the amount of force F per unit area A:
By the Action-Reaction principle, Pressure is the inward force per unit area that the container exerts on the fluid.
Pressure is the outward force per unit area that the fluid exerts on its container.
Atmospheric pressure comes from the weight of the column of air above us. At sea level, atmospheric pressure is (up to 10% lower during hurricanes!)
Pat = 1.01 105 N/m2
= 1.01 105Pa1 Pascal= 1 N/m2
= 14.7 lb/in2
= 1 bar (tire pressure gauges in Europe read 1, 2, …. bar)
The mass of the column of atmosphere above each square meter of the surface of the earth is
M = A P/g = [1m2] [1.01 105 N/m2 ] / [9.81 m /s2 ]
= 10.3 103 kg
This is huge, the mass of 1 m3 of water is only 103 kg
The density of air is about 1.0 kg/ m3.
The mass of a column of air of height h is M=rAh.
The equivalent height of the atmosphere is
h = (M/A)/r = [10.3 103 kg /m2]/[1.0 kg/ m3] 10 km
Actual height is >100 km because density decreases with height
Pressure in a fluid depends only on the depth h below the surface.
P = Pat + rgh r = density of fluid
Weight/Area of fluid
Weight/Area of atmosphere above fluid
IF the density of the fluid is constant and it has atmospheric pressure (Pat) at its surface.
Mass of fluid above depth h is
(density)(volume) = rhA
Force of gravity on fluid above depth h: W=rghA
To what depth in water must you dive to double the pressure exerted on your body?
P = Pat + rgh
rgh = Pat , h= Pat /rg
The variation in pressure at two different depths is given by:
P2 = P1 + rgh
When a hole is made in the side of a container holding water, water flows out and follows a parabolic trajectory. If the the container is dropped in free fall, the water flow
2. stops altogether.
3. goes out in a straight line.
4. curves upward.
A Force F1 is applied to area A1, displacing the fluid by a distance d1.
The pressure increase in the fluid is P=F1/A1.
The Pressure F1/A1 creates a force on the car F2= A2 (F1/A1).
The volume of fluid displaced on the left is V=d1 A1.
This equals the volume increase on the right V=d2A2.
Thus the work done by F1: W1 = F1d1 ,
is the same as the work done by the hydraulic system on the car:
W2=F2d2= d2(A2 F1/A1)=(d2A2 )(F1/A1)=( d1 A1)(F1/A1)= F1d1 = W1 :
Because the pressure in a fluid is greater below the object than above, there is an upward buoyant force Fb on any object in a fluid.
The upward buoyant force on an object is equal to the weight of the displaced fluid.
Nota bene: r is the density of the (displaced) fluid, not the density of the object (in green).
This result does not depend upon the shape of the immersed object.
F2/A = F1 /A + rgh
F2 = F1 + rghA
When an object floats, the magnitude of the upward buoyant force equals its weight. Therefore an object floats when it displaces an amount of fluid equal to its weight.
In order to float, an object must have a density less than or equal to that of the fluid in which it is immersed.
W=Mg = rblockVg, V = volume of block
F = fraction of block submerged
Volume displaced = fV
Weight of displaced water = rwaterfVg=Fb
Fb-W = Ma=0 (equilibrium)
Fb= W rwaterf Vg = rblockVg
f = rblock/rwater
How do steel ships float?
A boat is floating in a lake. The boat has a large rock in it. If the rock is thrown overboard, does the level of the water in the lake h increase, decrease or remain the same?
Inside the boat,
The rock displaces a volume of water equal in mass to the rock.
At the bottom of the lake, the rock displaces only a volume of water equal in mass to the rock. The density of the rock is about 4 times larger than the density of water. The height h of the water on the shore (not on the side of the boat) goes DOWN when you through the rock overboard.
A 0.12-kg balloon is filled with helium (density = 0.179 kg/m3). If the balloon is a sphere with a radius of 5.2 m, what is the maximum weight it can lift?
Density of air = 1.29 kg/m3.
If you have continuous flow of a fluid, then the rate of mass flow is the same at every point.
r1A1v1 = r2A2v2(general case: all liquids and gasses)
If the density does not change, which is true for most liquids:
A1v1 = A2v2 (liquids)
Conservation of energy in a flowing fluid leads to Bernoulli’s Equation: (work done by pressure = change in mechanical energy of a small volume of fluid)
P1 + ½rv12 + rgy1 = P2 + ½rv22 + rgy2
Here we assume that the density does not change.
Example: lift on an airplane wing
A horizontal pipe contains water at a pressure of 110 kPa flowing with a speed of 1.4 m/s. When the pipe narrows to one-half its original diameter, what is (a) the speed and (b) the pressure of the water?