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Basics We need to review fundamental information about physical properties and their units. These will lead us to two important methods: Conservation of Mass, and Conservation of Energy. We need to review fundamental information about physical properties and their units. These will lead us to two important methods: Conservation of Mass, and Conservation of Energy.

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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.

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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

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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

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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

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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, some data will be in English units. We must learn to use both. Système international d'unités pron system’ internah’tionana doo’neetay

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Data and Conversion Factors In your work 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 “

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Example Lava is flowing at a velocity of 30 meters per minute down Kilauea. What is this speed in feet per minute? 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 minute meter minute

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Chaining Conversion Factors Lava is flowing at a velocity of 30 meters per minute from a vent atop Kilauea. What is this speed in feet per second? 30 meters x feet x 1 minute = 1.64 feet minute meter 60 seconds sec

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Momentum (plural: momenta) Momentum (p) is the product of velocity and mass, p = mv In a collision between two particles, for example, if there is no frictional loss 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

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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 must have SI units of kg. m sec 2 1 kg. m is called a Newton (N) sec 2

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Statics If all forces and Torques are balanced, an object doesn’t move, and is said to be static Discussion Torques, See-saw F=2 F= F=3 The forces are balanced in the y direction force units (say, pounds) down are balanced by three pounds directed up. The torques are also balanced around the pivot: 1 pounds is 2 feet to the right of the pivot (= 2 foot-pounds) and 2 pounds one foot to the left = -2 foot - pounds

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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

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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

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A Conversion Factors Trick 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 (1 centimeter) 3 (1 meter) grams water = 1000 kg / m 3 Don’t forget to cube the 100cm

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Conservation of Mass – No Storage Mass flow rate Conservation of Mass : In a confined system “running full” and filled with an incompressible fluid, the same amount of mass that enters the system must also exit the system at the same time. 1 A 1 Vel 1 (mass inflow rate) = 2 A 2 Vel 2 ( mass outflow rate) What goes in, must come out. Notice all of the conditions/assumptions confined (pipe), running full, incompressible fluid (no compressible volatiles), same elevation (no Pressure differences). Volcanic pipe full of magma

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Mass Flow Rate for a vertical nozzle Lava is incompressible, so the density does not change and 1 = 2. The density cancels out, 1 A 1 V 1 = 2 A 2 V 2 so A 1 V 1 =A 2 V 2 Notice If A 2 V 1 Here A 2 < A 1.Thus lava exiting a smaller opening has a higher velocity than at inflow 1 A 1 V 1 (mass inflow rate) = 2 A 2 V 2 ( mass outflow rate) Consider lava flowing out an opening where the vent cross- sectional area is less than the magma chamber. A2A2 A1A1 V2V2 Just before the exit, assume (for now) P 2 = P 1, 2 = 1 exit V1V1

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Flow Rate For mass conservation with constant density, the flow rate is defined as Q = Velocity x Area Units are meter/sec x meters 2 Thus Q is Volume/time units m 3 /sec

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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 Distance has SI units of meters 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, Pv 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.

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Pressure Energy is Pressure x volume Energy has units kg. m 2 sec 2 So pressure energy must have the same units, and Pressure alone is kg. m sec 2 m 2 So if we multiply Pressure by a unit volume m3 we get units of energy m3m3

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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

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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

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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.

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Conservation of Energy We said earlier “Energy is Conserved” in a closed system. This means KE + PE + Pv + Heat = constant For simple systems involving fluids without friction heat losses, at two places 1 and 2 1/2 mV mgh 1 + P 1 v = 1/2 mV mgh 2 + P 2 v Usually we try to eliminate some of the terms. If both places are at the same pressure (say both touch the atmosphere) the pressure terms are identical

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A basaltic fountain on Kilauea volcano reaches a height of 53 Meters. What was the exit velocity at the vent? P 1 =P 2 = P atm =0 P atm =0 is a standard called gauge pressure At the vent (1) the height is zero, so there is only kinetic energy, KE = 1/2mV 2 At the top, h = 53 meters, the particles stop briefly before falling back to earth. There is only potential energy, Pot.E. = mgh, at (2). The masses are the same, so they cancel.

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Ex. 2 - Conservation of Energy Problem A tall volcano fills its neck with magma to an elevation of h 1 = 1000 m above a weak area. The pressure forms a rupture in the weak area. Define the height there as h 2 = 0 meters. How fast is the lateral blast? The rupture has area A 2 = m 2, small compared to the magma chamber surface with area A 1 = 3 x 10 7 m 2. Therefore assume V 1 ~ 0

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Step 1. Calculate Pressure at depth Pressure can be calculated as P = g h For an Andesitic magma with density = 2450 kg/m 3 If the surface of the magma chamber is at atmospheric pressure, what is the pressure 1000 meters below the surface? See the handout

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Step 2 Calculate the velocity Calculate V 3 using conservation of energy with the calculated pressure of magma P 2 entering the pipe and exiting to atmospheric Pressure P 3 via the rupture. V2 ~ 0, P 2 P 3 ~0 = P atm V3 = ?

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Specific Energy It will be convenient to separate out the mass. Since the “mass in” is the same as the “mass out”, we will be able to cancel them when necessary. The conservation of energy terms are:

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Handout We will walk through the steps in a handout, an then do similar problems in class and for homework.

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