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

Scalars and Vectors A scalar is a quantity with a size, for example mass or length A vector has a size (magnitude) and a direction.

Velocity 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 miles per hour, and a direction, South. The combination of these two properties, speed and direction, forms the vector quantity Velocity

Vector 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 & 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 = 180 o

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/sec 2

SI Units: Kilogram, 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/sec 2 Unfortunately, in Hydrology our clients are mostly civilians, who expect answers in English units. We must learn to use both. Système international d'unités pron dooneetay

Data 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 “ netpt.php?yr=2010&mo=12&dy=1&qc=&hr=10&element_id%5B%5D=24&states=NJ&newdc=1

Example 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 feet = feet second meter second

Flow Rate Q = V. A Flow Rate Q = V. A The product of velocity and area is a flow rate V [meters/sec] x A [meters 2 ] = Flow Rate [m 3 /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 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 5 2 = 78.54m 2 Q = VA = 30 m/s x m 2 Q = m 3 /s Radius r = D/2 = 5 m

Chaining Conversion Factors Water is flowing at a rate of 3000 meters cubed per second from a spillway outlet. What is this flow rate in feet 3 per hour? 3000 m 3 x 60 sec x 60 min = m 3 /hour sec min hour m 3 x (3.281 feet) 3 = ft 3 /hr hour ( 1 meter) 3

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 m 1 = m 2, one with initial speed v 1, the other at rest v 2 = 0, m 1 v 1 + m 2 v 2 = constant

Force Force is the change in momentum with respect to time. A normal speeds, Force is the product of Mass (kilograms) and Acceleration (meters/sec 2 ), so Force F = ma So Force must have SI units of kg. m sec 2 1 kg. m is called a Newton (N) sec 2 An example of Force is weight, F = mg

Statics 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-saw F=2 F= F=3 Both forces and torques balanced

Pressure Pressure is Force per unit Area So Pressure must have units of kg. m sec 2 m 2 1 kg. m is called a Pascal (Pa) sec 2 m 2

Density Density is the mass contained in a unit volume Thus density must have SI units kg/m 3 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 Factors  Suppose you need the density of water in kg/m 3. You may recall that 1 cubic centimeter (cm 3 ) of water has a mass of 1 gram.  1 gram water x (100 cm) 3 x 1 kilogram = 1000 kg / m 3  (centimeter) 3 (1 meter) grams  water = 1000 kg / m 3 Don’t forget to cube the 100

Mass Flow Rate Mass Flow Rate is the product of the Density and the Flow Rate i.e. Mass Flow Rate =  AV elocity Thus the units are kg m 2 m = kg/sec m 3 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.  1 A 1 V 1 (mass inflow rate) =  2 A 2 V 2 ( 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.

Energy 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. m 2 is called a N. m or Joule (J) sec 2 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 volume An isolated system, as contrasted with an open system, is a physical system that does not interact with its surroundings.

Kinetic Energy Kinetic Energy (KE) is the energy of motion KE = 1/2 mass. Velocity 2 = 1/2 mV 2 SI units for KE are 1/2. kg. m. m sec 2 Note 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 Energy s

Potential 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 sec 2 Note that the first two units make a Newton (force) and the remaining unit is meters, so the units of PE are indeed Energy Note also, these are the same units as for KE

KE 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. V 2 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 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 mV mgh 1 + P 1 v = 1/2 mV mgh 2 + P 2 v If both places are at the same pressure (say both touch the atmosphere) the pressure terms are identical 1/2 mV mgh 1 + P 1 v = 1/2 mV mgh 2 + P 2 v

Example 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 A 2 = 10 m 2, small compared to the lake surface with area A 1 = 3,000,000 m 2. Therefore assume V 1 ~ 0. Calculate V 2. note m 1 = m 2  Method: only PE at 1, KE at 2 mgh 1 =1/2mV 2 2 V 2 = 2gh 1/2 mV mgh 1 = 1/2 mV mgh m/sec