9.3 Density and Pressure

Density  and can depend on temperature. Specific gravity .



Pressure in a fluid   


At any particular depth, the pressure is constant throughout a fluid.




Worked Example: Problem #16

A 70-kg man in a 5.0 kg chair tilts back so that all the weight is balanced on two legs of the chair. Assume that each leg makes contact with the floor over a circular area with a radius of 1.0 cm, and find the pressure exerted on the floor by each leg.


9.4 Variation of Pressure with Depth






In equilibrium, all points at the same depth must be at the same pressure. Otherwise a net force would be applied and the fluid would accelerate.



Pick a volume of fluid a distance h below the surface:




P0 =1.013x105 Pa at sea level


P increases with depth by an amount mgh.


NOTE that an increase in pressure applied to an enclosed fluid is transmitted undiminished to every point in the fluid (including the walls of the container)- Pascal’s Principle


Worked Example:


At what depth in the ocean is the pressure twice that of the atmosphere alone (the density of seawater is about 1.02 kg m-3)?


What to wear at a depth of 200m?


Application: Hydraulic Press







Can you think of other applications of this principle?


9.5 Pressure Measurements


Two devices for measuring pressure:


open-tube manometer




Note that P = absolute (true) pressure inside the bulb. P-P0 is the gauge pressure, the pressure that is added to the atmospheric pressure to equal P.




1 atm.  pressure equal to a 0.76 m column of mercury at T=0°C and g=9.80665 m s-2.




You can read Torecelli’s own description of his barometer here.


Blood pressure






Blood pressure is measured in terms of the column of mercury (in millimeters) that could be supported by the pressure inside the arteries at two times: maximum thrust by the heart, and when the heart is relaxed. These are normally about 120 mm and 80 mm, respectively.


Recent medical guidelines suggest that the familiar old “normal” 120/80 values are too high, and that somewhat lower values are desired!







A collapsible plastic bag contains a glucose solution. If the average gauge pressure in the artery is 1.33x104 Pa, what must be the minimum height h of the bag in order to infuse glucose into the artery? Assume that the specific gravity of the solution is 1.02.




9.6 Buoyant Forces and Archimedes’s Principle


Any object completely or partially submerged in a fluid is buoyed up by a force whose magnitude is equal to the weight of fluid displaced by the object.




Sink or Float? Depends on B-w where w is the weight of the object.


For a completely submerged object, the volume of the object is the volume of displaced fluid,




For an object floating partially submerged on the surface,



Worked Example: Problem 26


A frog in a hemispherical pod finds that he just floats without sinking in a fluid of density 1.35 g/cm3. If the pod has a radius of 6.00 cm and negligible mass, what is the mass of the frog?


Worked Example: Problem 28


The density of ice is 920 kg/m3n and that of seawater is 1030 kg/m3. What fraction of the total volume of an iceberg is exposed?

9.7 Fluids in Motion


2 types of flow:  laminar (streamline) & turbulent


Example: Numerical simulation of flow over a racing car. Here, the pressure is color-coded, with blue being low pressure and red being high pressure. The flow lines are drawn in. Note that while the flow is laminar over much of the car, it breaks up into turbulent eddies behind the car.


For more neat simulations go here.


Ideal Fluids:

  1. nonviscous
  2. incompressible
  3. steady (does not depend on time)
  4. not turbulent


Equation of Continuity


For an incompressible fluid, flowing with no added “sources” or “sinks”:





Can you think of some examples of this principle?


Bernoulli’s Equation


Here, we will look at how the pressure changes in a laminar fluid flow.



Now, part of the work goes into changing the KE of the fluid, and part goes into changing the gravitational potential energy (mgh stuff).



Venturi Tube:




The increase in velocity of the fluid is accompanied by a drop in its pressure!


9.8 Other Applications


Aircraft Wing:


When air flows over the wing of an aircraft, the flow is faster over the more curved top than on the bottom, so that the pressure is lower on top than on the bottom. (Note: air is compressible, but the effect is small in this case and can be ignored). The tilt also aids lift. But turbulence disrupts the flow, diminishing the effect.




A stream of air passing over a tube dipped in a liquid causes the liquid to rise in the tube. Used in perfume atomizer bottles and paint sprayers.



Vascular Flutter:


The constriction in the blood vessel speeds up going through the constriction. The lower pressure causes the vessel to close, stopping the flow. Without flow, there is no Bernoulli effect, and blood pressure causes it to re-open. The process repeats.



Applying Physics to the Home

Consider the portion of a home plumbing system shown in the figure to the left. The water trap in the pipe below the sink captures a plug of water that prevents sewer gas from finding its way from the sewer pipe, up the sink drain, and into the home. Suppose the dishwasher is draining, so that water is moving to the left in the sewer pipe. What is the purpose of the vent, which is open to the air above the roof of the house? In which direction is air moving at the opening of the vent, upward or downward?


Worked Example


What is the net upward force on an airplane wing of area 20.0 m2 if the airflow is 300 m/s across the top of the wing and 280 m/s across the bottom?



Worked Example: Problem #43


A hypodermic syringe contains a medicine with the density of water. The barrel of the syringe has a cross-sectional area of 2.50x10-5 m2. In the absence of a force on the plunger, the pressure everywhere is 1.00 atm. A force F of magnitude 2.00 N is exerted on the plunger, making medicine squirt from the needle. Determine the medicine’s flow speed through the needle. Assume that the pressure in the needle remains equal to 1.00 atm and that the syringe is horizontal.









9.9 Surface Tension, Capillary Action, and Viscous Fluid Flow


Surface Tension


The combined electrical attraction of molecules in a fluid gives rise to a force that tends to minimize the surface area of the fluid. This makes raindrops spherical. If they weren’t we would not see rainbows the way we do!


This surface tension γ acts like a local force along the surface of the fluid:


Note that the units are the same as the spring constant.


The surface tension can support small objects placed on top of the surface (such as a needle, which will float if placed carefully on the surface of still water), and hold back others from leaving it (this impedes evaporation from a body of water, for example).


The surface tension of a fluid can be measured with a device like that shown here. If the force required to break free of the water is F, then


where r is the radius of the hoop. (Here we need to factor of 2 because the surface tension exerts forces on the inside and the outside of the ring.

The following table lists the surface tension of some common fluids.


Note that γ depends on temperature. At higher T, the molecules are not as tightly bound together. You can also alter the surface tension of fluids using additives.


Surfaces of Liquids


When water sits on a surface or in a container, the shape the water takes depends on whether it is more strongly attracted to itself (cohesion) or to the “other” material (adhesion).


Detergents  wet  allows water to penetrate clothes when washing and to spread over glass surfaces better.


Repellants  water beads up & penetrates less.


Capillary Action

Wetting  pulls up

Examples  paper towels, sponges, mops, finger-prick blood samples

Non-wetting  pushes down


Worked Example: Problem#56


A staining solution used in a microbiology laboratory has a surface tension of 0.088 N/m and a density 1.035 times that of water. What must be the diameter of a capillary tube so that this solution will rise to a height of 5 cm? (Assume a contact angle of zero).

Viscous Fluid Flow


Viscosity  the internal friction of a fluid. Resistance to shear stress.





Poiseuille’s Law






Affects blood flow, squeezing Krazy Glue gel out of its tube, etc.


Reynolds Number


When is the onset of turbulence? Fluid flow in a pipe of diameter d:





RN < 2000    laminar flow

2000 < RN < 3000 unstable…….

3000 < RN   turbulent flow



9.10 Transport Phenomena




Net movement of a population across a “cross-section” by random walk from a region where the concentration is higher to a region where it is lower.






Movement of water from a region where its concentration is high, across a selectively permeable membrane, into a region where it is lower.




Note use in artificial kidneys. Used in both hemodyalisis and paritoneal dialysis.


Motion through a Viscous Medium


Resistive force on spherical object of radius r:  

Terminal Speed  net force goes to zero  velocity is constant  


Sedimentation & Centrifugation - READ