BasicsBasics

• We need to review fundamental We need to review fundamental information about physical properties information about physical properties and their units.and their units.

Scalars and VectorsScalars and Vectors• A scalar is a quantity with a size, for

example mass or length

• A vector has a size (magnitude) and a direction.

http://www.engineeringtoolbox.com/average-velocity-d_1392.html

VelocityVelocity• Velocity is the rate and direction of

change in position of an object.

• For example, at the beginning of the Winter Break, our car had an average speed of 61.39 miles per hour, and a direction, South. The combination of these two properties, speed and direction, forms the vector quantity Velocity

http://www.engineeringtoolbox.com/average-velocity-d_1392.html

Vector ComponentsVector Components• Vectors can be broken down into components

• For example in two dimensions, we can define two mutually perpendicular axes in convenient directions, and then calculate the magnitude in each direction

• Vectors can be added

• The brown vector plus

the blue vector equals

the green vector

Trig & GeometryTrig & Geometry• cos = adj / hyp = abs /hyp• sine = opp / hyp = ord / hyp• tan = ord / abs• Usually angle known, solve • eqn. to find size of an unknown

The sum of angles inside a triangle = 180o

Vectors 2: Acceleration.Vectors 2: Acceleration.• Acceleration is the change in Velocity during

some small time interval. Notice that either speed or direction, or both, may change.

• For example, falling objects are accelerated by gravitational attraction, g. In English units, the speed of falling objects increases by about

g = 32.2 feet/second every second, written g = 32.2 ft/sec2

SI Units: SI Units: Kilogram, meter, secondKilogram, meter, second

• Most scientists and engineers try to avoid English units, preferring instead SI units. For example, in SI units, the speed of falling objects increases by about 9.81 meters/second every second, written

g = 9.81 m/sec2

• Unfortunately, in Hydrology our clients are mostly civilians, who expect answers in English units. We must learn to use both.

http://en.wikipedia.org/wiki/International_System_of_Units

Système international d'unités pron dooneetay

Data and Conversion FactorsData and Conversion Factors• In your work as a hydrologist, you will be

scrounging for data from many sources. It won’t always be in the units you want. We convert from one unit to another by using conversion factors.

• Conversion Factors involve multiplication by one, nothing changes

• 1 foot = 12 inches so 1 foot = 1 12 “

http://waterdata.usgs.gov/nj/nwis/current/?type=flow

http://climate.rutgers.edu/njwxnet/dataviewer-netpt.php?yr=2010&mo=12&dy=1&qc=&hr=10&element_id%5B%5D=24&states=NJ&newdc=1

ExampleExample

• Water is flowing at a velocity of 30 meters per second through a canyon. What is this speed in feet per second?

• Steps: (1) write down the value you have, then (2) select a conversion factor and write it as a fraction so the unit you want to get rid of is on the opposite side, and cancel. Then calculate.

• (1) (2)• 30 meters x 3.281 feet = 98.61 feet

second meter second

Flow Rate Q = V Flow Rate Q = V .. A A

• The product of velocity and area is a flow rate

• V [meters/sec] x A [meters2] = Flow Rate [m3/sec]

• Notice that flow rates have units of Volume/ second

• It is very important that you learn to recognize which units are correct for each measurement or property.

Example ProblemExample Problem

• Water is flowing at a velocity of 30 meters per second though a sea arch that has a diameter of 10 meters. What is the flow rate?

A = x 52 = 78.54m2

Q = VA = 30 m/s x 78.54 m2

Q = 2356.2 m3/s

Radius r = D/2 = 5 m

Chaining Conversion FactorsChaining Conversion Factors• Water is flowing at a rate of 3000 meters cubed per

second from a spillway outlet. What is this flow rate in feet3 per hour?

•

• 3000 m3 x 60 sec x 60 min = 10800000 m3/hour

sec min hour

10800000 m3 x (3.281 feet)3 = 381454240. ft3/hr

hour ( 1 meter) 3

Momentum (plural: momenta)Momentum (plural: momenta)• Momentum (p) is the product of velocity and mass, p =

mv• In a collision between two particles, for example, the total

momentum is conserved.

• Ex: two particles collide and m1 = m2, one with initial speed v1 ,

the other at rest v2 = 0,

• m1v1 + m2v2 = constant

ForceForce• Force is the change in momentum with respect to time.

• A normal speeds, Force is the product of Mass (kilograms) and Acceleration (meters/sec2), so Force F = ma

• So Force must have SI units of kg . m

sec2

• 1 kg . m is called a Newton (N)

sec2

An example of Force is weight, F = mg

Statics and DynamicsStatics and Dynamics

• If all forces and torques are balanced, an object doesn’t move, and is said to be static. We will use force balances shortly.

• Torque is force at some distance

• Demo Torques, ruler, See-sawF=2

F=1

-1 0 +2

F=3

Both forces and torques balanced

PressurePressure

• Pressure is Force per unit Area

• So Pressure must have units of kg . m

sec2 m2

• 1 kg . m is called a Pascal (Pa)

sec2 m2

DensityDensity• Density is the mass contained in a unit volume

• Thus density must have SI units kg/m3

• The symbol for density is pronounced “rho”

• Very important is not a p, it is an r• It is NOT the same as pressure

Chaining Conversion FactorsChaining Conversion Factors

Suppose you need the density of water in kg/m3. You may recall that 1 cubic centimeter (cm3) of water has a mass of 1 gram.

1 gram water x (100 cm)3 x 1 kilogram = 1000 kg / m3

(centimeter)3 (1 meter)3 1000 grams

water = 1000 kg / m3

Don’t forget to cube the 100

Mass Flow RateMass Flow Rate

• Mass Flow Rate is the product of the Density and the Flow Rate

• i.e. Mass Flow Rate = AVelocity

• Thus the units are kg m2 m = kg/sec

m3 sec

Conservation of Mass – No Storage

Conservation of Mass : In a confined system “running full” and filled with an incompressible fluid, all of the mass that enters the

system must also exit the system at the same time.

1A1V1(mass inflow rate) = 2A2V2( mass outflow rate)

What goes in, must come out.

Notice all of the conditions/assumptions confined (pipe), running full (no compressible air), horizontal (no Pressure differences) incompressible fluid.

EnergyEnergy• Energy is the ability to do work, and work and

energy have the same units• Work is the product of Force times distance, • W = Fd

• 1 kg . m2 is called a N.m or Joule (J)

sec2

• Energy in an isolated system is conserved • KE + PE + Pv + Heat = constant

N.m is pronounced Newton meter, Joule sounds like Jewel. KE is Kinetic Energy, PE is Potential Energy, P-v is Pressure energy, v is unit volumeAn isolated system, as contrasted with an open system, is a physical system that does not interact with its surroundings.

Kinetic EnergyKinetic Energy

• Kinetic Energy (KE) is the energy of motion

• KE = 1/2 mass . Velocity 2 = 1/2 mV2

• SI units for KE are 1/2 . kg . m . m• sec2Note the use of m both for meters and for mass. The context will tell you which.

That’s the reason we study units.Note that the first two units make a Newton (force) and the remaining unit is meters, so the units of KE are indeed Energys

Potential EnergyPotential Energy

• Potential energy (PE) is the energy possible if an object is released within an acceleration field, for example above a solid surface in a gravitational field.

• The PE of an object at height h is

PE = mgh Units are kg . m . m

sec2

Note that the first two units make a Newton (force) and the remaining unit is meters, so the units of PE are indeed EnergyNote also, these are the same units as for KE

KE and PE exchangeKE and PE exchange• An object falling under gravity loses

Potential Energy and gains Kinetic Energy.• A pendulum in a vacuum has potential

energy PE = mgh at the highest points, and no kinetic energy because it stops

• A pendulum in a vacuum has kinetic energy KE = 1/2 mass.V2 at the lowest point h = 0, and no potential energy.

• The two energy extremes are equal

Stops v=0 at high point, fastest but h = 0 at low point.Without friction, the kinetic energy at the lowest spot (1) equals the potential energy at the highest spot, and the pendulum will run forever.

Conservation of EnergyConservation of Energy• We said earlier “Energy is Conserved” • This means

KE + PE + Pv + Heat = constant • For simple systems involving liquid water without

friction heat, at two places 1 and 2

1/2 mV12 + mgh1 + P1v = 1/2 mV2

2 + mgh2 + P2v

If both places are at the same pressure (say both touch the atmosphere) the pressure terms are identical

• 1/2 mV12 + mgh1 + P1v = 1/2 mV2

2 + mgh2 + P2v

Example ProblemExample Problem• A Watchung Lava flow dammed a

proglacial lake, Lake Passaic, south of the melting Wisconsinan glacier. A leaky area had an opening h = 100 m below the water level. The opening had an area A2 = 10 m2 , small compared to the lake surface with area A1 = 3,000,000 m2. Therefore

assume V1 ~ 0.

• Calculate V2. note m1 = m2

Method: only PE at 1, KE at 2 mgh1=1/2mV2

2 V2 = 2gh

1/2mV12 + mgh1 = 1/2mV2

2 + mgh2

44.29m/sec