Basics

1. 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.

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

3. http://www.engineeringtoolbox.com/average-velocity-d_1392.htmlhttp://www.engineeringtoolbox.com/average-velocity-d_1392.html 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 61.39 miles per hour, and a direction, South. The combination of these two properties, speed and direction, forms the vector quantity Velocity

4. 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

5. 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

6. 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/sec2 • Unfortunately, some data will be in English units. We must learn to use both. Système international d'unités pron system’ internah’tionana doo’neetay http://en.wikipedia.org/wiki/International_System_of_Units

7. 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 “

8. 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 3.281 feet = 98.61 feet minute meter minute

9. 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 3.281 feet x 1 minute = 1.64 feet minute meter 60 seconds sec

10. 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 m1 = m2, one with initial speed v1 , the other at rest v2 = 0, • m1v1 + m2v2 = constant

11. Force • 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 must have SI units of kg . m sec2 • 1kg . m is called a Newton (N) sec2

12. Statics • If all forces and Torques are balanced, an object doesn’t move, and is said to be static • Discussion Torques, See-saw The forces are balanced in the y direction. 2 + 1 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 F=2 F=1 -1 0 +2 F=3

13. Pressure • 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

14. Density • Density is the mass contained in a unit volume • Thus density must have SI units kg/m3 • The symbol for density is r, pronounced “rho” • Very important r is not a p, it is an r • It is NOT the same as pressure

15. A Conversion Factors Trick • 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)3x 1 kilogram = 1000 kg / m3 • (1 centimeter)3 (1 meter)3 1000 grams • rwater=1000 kg / m3 Don’t forget to cube the 100cm

16. 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. r1A1Vel1(mass inflow rate) = r2A2Vel2( mass outflow rate) Volcanic pipe full of magma 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).

17. Mass Flow Rate for a vertical nozzle Consider lava flowing out an opening where the vent cross-sectional area is less than the magma chamber. r1A1V1(mass inflow rate) = r2A2V2( mass outflow rate) Lava is incompressible, so the density does not change and r1= r2. The density cancels out, r1A1V1 = r2A2V2 so A1V1 =A2V2 Notice If A2 < A1 then V2 >V1 Here A2 < A1 .Thus lava exiting a smaller opening has a higher velocity than at inflow exit V2 A2 V1 A1 Just before the exit, assume (for now) P2 = P1, r2 = r1

18. 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 m3/sec

19. 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 • 1kg . 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, 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.

20. Pressure Energy is Pressure x volume • Energy has units kg . m2 sec2 So pressure energy must have the same units, and Pressure alone is kg . m sec2 m2 So if we multiplyPressureby a unit volume m3 we get unitsof energy m3

21. Kinetic 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 • sec2 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

22. 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 sec2 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

23. 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.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.

24. 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 mV12 + mgh1 + P1v = 1/2 mV22 + mgh2 + P2v 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

25. A basaltic fountain on Kilauea volcano reaches a height of 53 Meters. What was the exit velocity at the vent? P1 =P2 = Patm =0 At the vent (1) the height is zero, so there is only kinetic energy, KE = 1/2mV2 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. Patm =0 is a standard called gauge pressure

26. Ex. 2 - Conservation of Energy Problem • A tall volcano fills its neck with magma to an elevation of h1 = 1000 m above a weak area. The pressure forms a rupture in the weak area. Define the height there as h2 = 0 meters. • How fast is the lateral blast? • The rupture has area A2 = 10000 m2 , small compared to the magma chamber surface with area A1 = 3 x 107 m2. Therefore assume V1 ~ 0

27. Step 1.Calculate Pressure at depth • Pressure can be calculated as P = rgDh • For an Andesitic magma with density r = 2450 kg/m3 • If the surface of the magma chamber is at atmospheric pressure, what is the pressure 1000 meters below the surface? See the handout

28. Step 2Calculate the velocity • Calculate V3 using conservation of energy with the calculated pressure of magma P2 entering the pipe and exiting to atmospheric Pressure P3 via the rupture. V3 = ? V2 ~ 0, P2 P3 ~0 = P atm

29. 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:

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