exam 3 lec m
TRANSCRIPT
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Exam 3 Lecture
Conservation Laws
Energy & Momentum
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Definitions
Energy the ability to do work; a scalarquantity associated with the state of one ormore objects
Kinetic energy energy associated with themotion of a particle
Potential energy energy associated withthe position of a particle
Work energy transfer to or from an object
Joule the unit of energy 1J=1Nm
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Types of Energy
Mechanical energy includes PE and KE
Chemical energy important in biology
Electromagnetic energy important in
industry
Thermal energy heat
Nuclear energy energy of atoms
Energy can be converted from one form to another
but never created or destroyed conservation of
energy
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Some Formulas
Kinetic Energy
Work
Work with constant
force
Work of gravity
Work of spring
2
21 mvK !
y!f
r
r xdFW 0TT
UcosFddFW !y!TT
mghW !
221 kxW !
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The work a force does on an object depends on
the angle between the force and the distance the
object moves
090cos !! NdWN
090cos !! dFWgg
40cos22dFW !
30cos11dFW !
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The angle between
the force and the
distance the object
moves is important
for the sign of the
work done also If angle is less than
90 work is positive
If angle is greaterthan 90 work is
negative
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Work Kinetic Energy Theorem
Work is an energy transfer from one form to
another
Work Kinetic Energy theorem (when
energy is transferred to motion)
Sign corresponds to where energy
transferred
ifnet KKW !
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Energy transferred to
the system corresponds
to positive work and
increase in kinetic
energy
Energy transferred from
the system corresponds
to negative work and adecrease in kinetic
energy
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For a variable force,
or work is the area under the curve graphically.
Remember area formulas from geometry
y! frr xdFW0
TT
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Power
Power the rate of doing work
Watt the unit of power
sJW 11 !
dt
dWPinst !t
EPave (
(!
vFPTT
y!
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Potential Energy
Potential energy (U) energy associatedwith position and conservative forces (also
considered stored energy)
Conservative forces a force which allows
two way conversion between K and U
Nonconservative (dissipative) forces force
in which energy is lost to K and U
Important to realize that the choice of a
reference point is arbitrary (because path
independent)
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Properties of work for conservative forces
It can always be expressed as difference between initial
and final values of a potential energy function It is reversible
It is independent of path and depends only on initialand final points
When initial and final points are the same work is zero
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Potential Energy cont.
Conservative force potential energy
Gravitational potential energy
Elastic potential energy
ymgU (!(
222
1if xxkU !(
y!( fi
x
xxdFUTT
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Add up area
under curve of
Force vs.
Distance graph
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Conservation of Mechanical Energy
Mechanical energy is sum of potential and
kinetic energy of a system
If only conservative forces do work in a
system mechanical energy is conserved
Important because problems can be solved
without knowing forces acting on system only energy considerations are important
1122 UKUK !
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Conservation of mechanical energy
problems benefit from looking at the
problems in sections
CBAiMEMEMEME !!!
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Problem Solving Strategy
Look at the problem and determine allforces acting in the system
Split the problem into parts using breaks at
points where all the energy can be quantized
Determine all the energy forms present at
each part
Total energy of each part is equal
Solve for missing energy or variables
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Potential Energy Curves Turning point a point where the particle motion
reverses
Equilibrium a point where the particle does not
move
Stable equilibrium any movement away from thispoint results in a force back to it (U is minimum)
Unstable equilibrium any movement away from this
point results in a force away from it (U is maximum)
Neutral equilibrium any displacement away from
equilibrium results in no force (U is constant)
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More about Potential Energy Curves
We can find the force a particle feels at a
given position on a potential energy curve
F is the negative slope of the tangent
We can also determine what the kinetic
energy of the system is if we know the total
mechanical energy
dU dU dU
F i j k Udx dy dz
! !
r
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Conservation ofTotal Energy
Conservation of total energy is true always(we just might know where some of the
energy has gone)
The total energy of a system can onlychange by amounts of energy transferred to
or from the system
intmechanical t hermal ernal external
gi si i other gf sf f
E E E E E
U U K W U U K
( ! ( ( ( ( !
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In conservation of mechanical energy,energy loss due to friction is ignored
In conservation of total energy, all energylosses are considered
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Satellites
First consider energy conservation realizing
gravity is a conservative force
To determine the escape velocity from a mass use
the total energy =0
r
GMv
2!
r
Mm
r
Mm
r
Mm
r
MmmvUKEtotal
22
1
21 2
!!
!!
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Rotational Energy Formulas
UXU cos!W
221 [IK!
UcosFdW !
2
21 mvK!
Translational Rotational
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When you find the kinetic energy for rolling you mustuse 2 terms, one for the translation motion and one forthe rotation motion
For problems you need to draw vector diagrams toanalyze the motion, but utilize torque for force
22
21
21
comcomroll MvIK ! [
EX I!
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Conservation of
energy still applies
as well asconservation of
momentum
ghv
mvmgh
2
21 2
!
!
Mm
ghmv
vMmmv
f
f
!
!
2
2
2
2
3
21
Mm
hmh
hgMmI
!d
d![
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Newtons 2nd Law for a System of
Particles332211 rmrmrmrmrM iicomTTTTT
!!
332211 vmvmvmvmvM iicomTTTTT
!!
332211 amamamamaM iicomTTTTT
!!
netcom FFFFaM
TTTTT
!! 321
The motion of the center of mass depends only
on external forces. Therefore the forces
summed to make the resultant force in theprevious equation are the external forces acting
on the system of particles.
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Linear Momentum
Linear Momentum the product of themass and the velocity of a body or system
of particles
It is a term for describing objects in motion
and can be related to the net force acting on
a body or system of particles
vmp
TT
!
net
dpF
dt!
rr
comv
M
TT
!
net
dPF
dt!
rr
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Impulse
Collision an isolated event in which two
or more bodies exert relatively strong forces
on each other for a relatively short time
Impulse a measure of the strength and
duration of the collision force
if pppJTTTT
!(!
! fi
t
tdttFJ
TT
tFJavg(!
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The impulse J is
equal to the areaunder the curve of a
Force vs. time graph
for a variable force
The impulse is also
equal the area under
the rectangle of
height Favg
From Newtons 2nd
law
(
!!
t
vvmamF
avg0
TTTT
0vmvmtFTTT
!(
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Conservation of Linear Momentum Conservation of linear momentum if the
net external force acting on a system is 0,then the linear momentum of the systemdoesnt change
fi PPTT
! ffffiiii vmvmvmvm 22112211TTTT
!
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Angular Momentum
Angular momentum is
Angular momentum of a rigid body rotating about a
fixed axis
vrmprL TTTTT
v!v!
[IL !
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Angular momentum
depends on the mass ofthe object, the velocity of
the object and the
distance of the object
from the axis of rotation
Torque depends on the
force and the distance of
the object from the axisof rotation
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Angular Momentum cont.
Angular momentum is conserved just like
linear momentum if the summation of the
external torques are zero
Therefore we can have rotating collisions
where angular momentum is conserved
dtLd
ext
TT
!! 0X fi LLTT
!
ffii IIII 22112211 [[[[ !
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As the moment of inertia changes so does the
angular velocity since angular momentum is
conserved
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dt
dvMv
dt
dMT
MaRvT
rel
rel
!!
!!
!f
irelifM
Mvvv ln
Vrel is the velocity of the exhaust with respect to
the rocket = vex in the figure
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Types of Collisions
There are three types of collisions:1. Purely elastic collisions the kinetic
energy of the system is conserved
2. Purely inelastic collisions the
objects which collided stick together3. Partially inelastic collisions the
kinetic energy of the system is not
conserved
We will consider collisions in 1Dand 2D
Note that in all collisions linear
momentum is conserved
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Conservation of Momentum in 1D
Both initially moving
Purely inelastic (both
final velocities are the
same)
m2
is initially at rest
Purely inelastic
ffii vmvmvmvm 22112211 !
ffivmvmvm
221111
!
vmmvmvm ii 212211 !
vmmvm i 2111 !
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Completely elastic collisions in 1D
Completely elastic
both objects initially
moving
1 2 21 1 2
1 2 1 2
2f i i
m m mv v v
m m m m
!
1 2 12 1 2
1 2 1 2
2f i im m mv v vm m m m
!
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Collisions in 1D cont.
Purely elastic collision if 2nd object isinitially at rest
If
If
If
if v
mm
mmv 1
21
211
! if v
mm
mv 1
21
12
2
!
21 mm ! 01 !fv if vv 12 !
21 mm ""
21 mm
if vv 11 }
if vv 11 }
if vv 12 2}
02
1
2
12 }
} if v
m
mv
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Collisions in 2D
For 2D collisions we must use vector component
notation to solve these problems Remember to treat the vector equations like 2
equations: x components and y components
Use the equations for conservation ofmomentum
For elastic collisions use the equation forconservation of kinetic energy
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Definitions Fluids a collection of molecules that are
randomly arranged and held together byweak cohesive forces and forces exerted bythe walls of a container
Density mass per volume Pressure force per area
Absolute pressure the total pressure at a
given depth Gauge pressure the difference between
absolute pressure and atmospheric pressure
V
m
(
(!V
A
F
(
(!
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More Definitions
Barometer a device used to measure
pressure of the atmosphere
Open-tube manometer a device used to
measure gauge pressure
Buoyant force the upward force on an
object in a fluid
Ideal fluid a fluid with 4 characteristicsallowing simple mathematical handling
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Fluids
Fluids tend to flow
The pressure at a point in a fluid in static
equilibrium depends on the depth of that point
but not on any horizontal dimension of the
fluid or its container
ghPP V!0
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The pressure at
all points at agiven depth is
the same
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Archimedes Principle
Any body
completely or
partially
submerged in a
fluid is buoyed
up by a force
equal to the
weight of the
fluid displaced
by the body
VgWFfluidb
V!!
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Pascals Law A change in pressure
applied to an enclosedfluid is transmittedundiminished to everypoint of the liquid and to
the walls of the container
The volume change istransmitted undiminishedalso
2
2
1
1
21
A
F
A
F!
!
2211
21
dAdA
VV
!
!
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Some DefinitionsSteady or laminar flow each particle of the fluid
follows a smooth path. Paths of differentparticles never cross each other. Velocity of the
fluid at any point remains constant in time.
Nonsteady or turbulent flow irregular flowcharacterized by small whirlpool-like regions
Viscosity characterizes the degree of internal
friction in the fluid
Streamline path taken by a fluid particle under
steady flow (velocity of fluid particle is always
tangent to the streamline)
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Ideal Fluid
Four characteristics:1. Steady flow
2. Incompressible fluid density constant in
time
3. Nonviscous fluid internal friction neglected
4. Irrotational flow no angular momentum of
fluid about any point
Use ideal fluids to discuss fluids in motion
because it is much simpler mathematically
and still provides useful results
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Some Equations
Equation of continuity for anincompressible fluid (Assumes volume per
time is constant)
2211 vAvARvol !!
tcons
AvRRvolmass
tan!
!! VV
Volume flow rate =
Rvol
Mass flow rate = Rmass
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Bernoullis equation
(Derived from
conservation of energy)
tconsgyvP tan
2
1 2 ! VV
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Lift
Bernoullis equation explains lift on anobject moving through a fluid
Factors influencing lift are:
1. Shape of object2. Its orientation with respect to fluid flow
3. Spinning motion
4.T
exture of objects surface
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Viscosity Viscosity is the internal friction of a liquid
A viscous fluid tend to stick to surfaces
Velocity is largest at the center of pipes