Transcript 1 Review Part 1 units n

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.
http://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.
http://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
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
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: 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
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 “
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
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
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
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
• 1 kg . m
sec2
is called a Newton (N)
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
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=1
-1
0
F=3
+2
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
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
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)3 x 1 kilogram = 1000 kg / m3
(1 centimeter)3
(1 meter)3
1000 grams
r water = 1000 kg / m3
Don’t forget to cube the 100cm
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).
Mass Flow Rate for a vertical nozzle
Consider lava flowing out an opening where the vent crosssectional 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
exit
V2
A2
so A1V1 =A2V2
Notice If A2 < A1 then V2 > V1
V1
A1
Here A2 < A1 .Thus lava exiting a
smaller opening has a higher
velocity than at inflow
Just before the exit, assume (for now) P2 = P1, r2 = r1
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
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 . 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.
Pressure Energy is
Pressure x volume
• Energy has
units kg . m2
m3
sec2
So pressure energy must have the same units,
and Pressure alone is
kg . m
sec2 m2
So if we multiply Pressure by a unit volume m3 we
get units of energy
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
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
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.
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
• 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
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
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
Step 2
Calculate 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
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:
Handout
• We will walk through the steps in a
handout, an then do similar problems in
class and for homework.