College Physics I: BIIG problem-solving method

Created Feb. 7, 2024 by C N Hiremath

( Chapter 11 )

Fluid Statics

  • A fluid is a state of matter that yields to sideways or shearing forces. Liquids and gases are both fluids.

  • The major distinction is that gases are easily compressed, whereas liquids are not.

  • Fluid statics is the physics of stationary fluids.

 

Matter

  • Matter most commonly exists as solid, liquid, or gas. Solids have a definite shape and a specific volume. Liquids have a definite volume but their shape changes depending on the container in which they are held. Gases have neither a definite shape or a specific volume.

 

Variation of Pressure with Depth

  • The pressure due to the weight of a fluid is

                P  =  h ρ g 

  • Problem (E11.3):  The dam is 500 m wide and the water is 80.0 m deep at the dam. The density of water is 1.0000 x 103 kg/m3.  Calculate the force against the dam.                                                                                                                   ( 3 x 1010  N )

 

Pascal’s Principle

  • A change in pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid and to the walls of its container.

  • The total pressure in a fluid is the sum of the pressures from different sources.

A hydraulic system is an enclosed fluid system used to exert forces.

                P1  =  P2                  F1 / A1  =  F2  / A2

  • Hydraulic systems can increase or decrease the force applied to them. They are analogous to simple levers. The most common hydraulic systems are power brakes.

  • Conservation of energy applied to a hydraulic system tells us that the system cannot do more work than is done on it.

 

Gauge Pressure & Absolute Pressure

  • Gauge pressure is the pressure relative to atmospheric pressure. It is positive for pressures above atmospheric pressure, and negative for pressures below it.

  • Absolute pressure is the sum of gauge pressure and atmospheric pressure.

                Pabs  =  Pg  +  Patm

  • There are a host of devices for measuring pressure.For example, tire gauges to blood pressure cuffs.

  • Problem (E11.7): Intravenous infusions are usually made with the help of the gravitational force. Assuming that the density of the fluid being administered is 1.00 g/ml, at what height should the IV bag be placed above the entry point so that the fluid just enters the vein if the blood pressure in the vein is 18 mm Hg above atmospheric pressure? Assume that the IV bag is collapsible. Use 1.0 mm Hg = 133 Pa.                                                                                                                        ( 0.24  m ) 

  • A barometer is a device that measures atmospheric pressure. The barometer can also be used as an altimeter, since average atmospheric pressure varies with altitude.

  • Conversion factors

1.0    atm  =  1.013 x 105  N/m2   =  101.3  kPa                                  1.0 mm Hg  =  133  Pa

 

Buoyant Force

  • Buoyant force is the net upward force on any object in any fluid.

  • If the buoyant force is less than the object’s weight, the object will sink. If the buoyant force is greater than the object’s weight,  The object will rise to the surface and float.

 

Archimedes’ Principle

  • Archimedes’ principle states that the buoyant force on an object equals the weight of the fluid it displaces.

                        FB  =  wfl

  • Specific gravity is the ratio of the density of an object to a fluid (usually water).

  • Problem (E11.8):  What is the maximum buoyant force that water could exert on a 1.00×107 kg solid steel if it were shaped into a boat that could displace 1.00×105 m3 of water? Compare this with the steel’s weight.          ( 9.80 x 108  N ) 

Cohesion and Adhesion in Liquids

  • Attractive forces between molecules of the same type are called cohesive forces.

  • Attractive forces between molecules of different types are called adhesive forces.

 

 ( Chapter 12 )

Fluid dynamics

  • Is the physics of fluids in motion. Examples - A column of smoke rises from a camp fire, water streams from a fire hose, blood courses through your veins.

  • Understand how does a nozzle increase the speed of water emerging from a hose?

 

Flow Rate

  • Flow rate Q is defined to be the volume V of fluid passing by some location through an area during a period of time t.

                Q  =  V / t

The SI unit for flow rate is m3/s.

  • In terms of average velocity v, flow rate is

                Q  =  V / t  =  A d / t  =  A ( d / t )   =  A v

where A is the cross-sectional area.

 

Equation of Continuity

  • For a incompressible fluid, the same amount of fluid must flow past any point in the tube in a given time to ensure continuity of flow. For points 1 and 2,

                Q1  =  Q2                                                 A1 v1 =  A2 v2

This is called the equation of continuity.

  • Since liquids are essentially incompressible, the equation of continuity is valid for all liquids. However, gases are compressible!

  • Problem (E12.2):  A nozzle with a radius of 0.250 cm is attached to a garden hose with a radius of 0.900 cm. The flow rate through hose and nozzle is 0.500 L/s. Calculate the speed of the water in the nozzle.                                                  ( 25.5  m/s ) 

 

Bernoulli’s Equation

  • Daniel Bernoulli (1700-1782), Swiss scientist, described the relationship between pressure and velocity in fluids.

  • Bernoulli’s equation states that for an incompressible, frictionless fluid, the following sum is constant:

                P  + (½) ρ v 2  +  ρ g h  =  constant

where, P is the absolute pressure, ρ is the fluid density, v is the velocity of the fluid, h is the height above some reference point, and g is the acceleration due to gravity.

  • If we follow a small volume of fluid along its path, various quantities in the sum may change, but the total remains constant.

  • Let the subscripts 1 and 2 refer to any two points along the path that the bit of fluid follows; Bernoulli’s equation becomes

                P1  + (½) ρ v1 2  +  ρ g h1  =   P2  + (½) ρ v2 2  +  ρ g h2

Bernoulli’s equation is a form of the conservation of energy principle.

Note that the second and third terms are the kinetic and potential energy.

  • Note, power is defined as the rate of energy transferred.  The SI unit is Watt (W).

                E / t  =  P Q                            

  • Problem (E12.4): The speed of water in a hose increases from 1.96 m/s to 25.5 m/s going from the hose to the nozzle. Calculate the pressure in the hose, given that the absolute pressure in the nozzle is 1.01×105 N/m2 (atmospheric, as it must be) and assuming level, frictionless flow.                                                                                                                    ( 4.24×105  N/m2 ) 

·         The Bernoulli principle helps explain lift generated by a wing, and thrust generated by the sails.

·         It is simply a kinematic equation for any object falling a distance h with negligible resistance.

In fluids, it is called Torricelli’s theorem.

 

Viscosity

  • Ideal fluids have little or no viscosity. Pouring a glass of juice!

  • Viscosity is due to fluid friction, and it affects the rate of fluid flow.Pouring honey!

  • Fluid viscosity is given by

                η  =  F L / v A

The SI units for viscosity are Ns/m2.

Laminar Flow

  • Laminar flow is characterized by the smooth flow of the fluid in layers that do not mix.

  • Turbulent flow, or turbulence, is characterized by eddies and swirls that mix layers of fluid together.

  • Streamlines are smooth and continuous when flow is laminar, but break up and mix when flow is turbulent.

 

Resistance to Laminar Flow

  • J. L. Poiseuille (1799–1869) French scientist derived an equation to understand the flow of blood, a turbulent fluid.

  • Poiseuille’s law for resistance: The resistance R to laminar flow of an incompressible fluid having viscosity η through a horizontal tube of uniform radius r and length l, is given by

                R  =  8 η l  /  π r 4

  • Note, r is raised to the fourth power, means that any change in the radius of a tube has a very large effect on resistance.

 

Poiseuille’s Law

  • Pressure difference causes the flow.  Flow rate Q is in the direction from high to low pressure. The greater the pressure differential between two points, the greater the flow rate. This relationship can be stated as

                Q  =  ( P2 − P1 ) / R                 

  • Poiseuille’s law for laminar flow is

                        Q  =  ( P2 − P1 ) π r 4 /  8 η l

Applies to an incompressible fluid.

  • Problem (E12.8): An intravenous (IV) system is supplying saline solution to a patient at the rate of 0.120 cm3/s through a needle of radius 0.150 mm and length 2.50 cm. What pressure is needed at the entrance of the needle to cause this flow, assuming the viscosity of the saline solution to be 1.002 x 10-3 Ns/m3?  The gauge pressure of the blood in the patient’s vein is 8.00 mm Hg.  Use 1 mm of Hg = 133 N/m2.                                                                                             ( 1.62 x 10 4  N/m2 ) 

 

The Onset of Turbulence

  • The Reynolds number NR can reveal whether flow is laminar or turbulent.

                NR  =  2 ρ v r / η

Where, ρ is the fluid density, v its speed, η its viscosity, and r the tube radius.

Reynolds number is a unitless quantity.

  • Laminar -NR below about 2000.  Turbulent - NR above about 3000.  Unstable flow - NR between 2000-3000.

  • Problem (E12.9):  An intravenous (IV) system is supplying saline solution to a patient at the rate of 0.120 cm3/s through a needle of radius 0.150 mm and length 2.50 cm. Calculate the Reynolds number for flow in the needle to verify the assumption that the flow is laminar, assuming the viscosity of the saline solution to be 1.002 x 10-3 Ns/m3. Also assume that the density of the saline solution is 1025 kg/m3.                                                                                                    ( 522 ) 

 

Diffusion, Osmosis, Dialysis, and Active Transport

  • Diffusion is the movement of substances due to random thermal molecular motion. It is slow process over macroscopic distances.  The densities of common materials are great enough that molecules cannot travel very far before having a collision that can scatter them in any direction, including straight backward.

  • Osmosis is the transport of water through a semipermeable membrane from a region of high concentration to a region of low concentration.

  • Dialysis is the transport of any other molecule through a semipermeable membrane due to its concentration difference.  Both processes can be reversed by back pressure.

  • Active transport is a process in which a living membrane expends energy to move substances across it. Reverse osmosis and reverse dialysis (also called filtration) are processes that occur when back pressure is sufficient to reverse the normal direction of substances through membranes. The kidneys, for example, not only use osmosis and dialysis — they also employ significant active transport to move substances into and out of blood. 


BIIG: Problems & Solutions


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