8/8/2019 exam 3 lec m

1/58

Exam 3 Lecture

Conservation Laws

Energy & Momentum

8/8/2019 exam 3 lec m

2/58

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

8/8/2019 exam 3 lec m

3/58

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

8/8/2019 exam 3 lec m

4/58

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 !

8/8/2019 exam 3 lec m

5/58

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 !

8/8/2019 exam 3 lec m

6/58

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

8/8/2019 exam 3 lec m

7/58

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 !

8/8/2019 exam 3 lec m

8/58

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

8/8/2019 exam 3 lec m

9/58

For a variable force,

or work is the area under the curve graphically.

Remember area formulas from geometry

y! frr xdFW0

TT

8/8/2019 exam 3 lec m

10/58

Power

Power the rate of doing work

Watt the unit of power

sJW 11 !

dt

dWPinst !t

EPave (

(!

vFPTT

y!

8/8/2019 exam 3 lec m

11/58

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)

8/8/2019 exam 3 lec m

12/58

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

8/8/2019 exam 3 lec m

13/58

Potential Energy cont.

Conservative force potential energy

Gravitational potential energy

Elastic potential energy

ymgU (!(

222

1if xxkU !(

y!( fi

x

xxdFUTT

8/8/2019 exam 3 lec m

14/58

Add up area

under curve of

Force vs.

Distance graph

8/8/2019 exam 3 lec m

15/58

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 !

8/8/2019 exam 3 lec m

16/58

Conservation of mechanical energy

problems benefit from looking at the

problems in sections

CBAiMEMEMEME !!!

8/8/2019 exam 3 lec m

17/58

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

8/8/2019 exam 3 lec m

18/58

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)

8/8/2019 exam 3 lec m

19/58

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

8/8/2019 exam 3 lec m

20/58

8/8/2019 exam 3 lec m

21/58

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

( ! ( ( ( ( !

8/8/2019 exam 3 lec m

22/58

In conservation of mechanical energy,energy loss due to friction is ignored

In conservation of total energy, all energylosses are considered

8/8/2019 exam 3 lec m

23/58

8/8/2019 exam 3 lec m

24/58

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

!!

!!

8/8/2019 exam 3 lec m

25/58

Rotational Energy Formulas

UXU cos!W

221 [IK!

UcosFdW !

2

21 mvK!

Translational Rotational

8/8/2019 exam 3 lec m

26/58

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!

8/8/2019 exam 3 lec m

27/58

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![

8/8/2019 exam 3 lec m

28/58

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.

8/8/2019 exam 3 lec m

29/58

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

8/8/2019 exam 3 lec m

30/58

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(!

8/8/2019 exam 3 lec m

31/58

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

!(

8/8/2019 exam 3 lec m

32/58

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

!

8/8/2019 exam 3 lec m

33/58

Angular Momentum

Angular momentum is

Angular momentum of a rigid body rotating about a

fixed axis

vrmprL TTTTT

v!v!

[IL !

8/8/2019 exam 3 lec m

34/58

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

8/8/2019 exam 3 lec m

35/58

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 [[[[ !

8/8/2019 exam 3 lec m

36/58

As the moment of inertia changes so does the

angular velocity since angular momentum is

conserved

8/8/2019 exam 3 lec m

37/58

8/8/2019 exam 3 lec m

38/58

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

8/8/2019 exam 3 lec m

39/58

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

8/8/2019 exam 3 lec m

40/58

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 !

8/8/2019 exam 3 lec m

41/58

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

!

8/8/2019 exam 3 lec m

42/58

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

8/8/2019 exam 3 lec m

43/58

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

8/8/2019 exam 3 lec m

44/58

8/8/2019 exam 3 lec m

45/58

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

(

(!

8/8/2019 exam 3 lec m

46/58

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

8/8/2019 exam 3 lec m

47/58

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

8/8/2019 exam 3 lec m

48/58

The pressure at

all points at agiven depth is

the same

8/8/2019 exam 3 lec m

49/58

8/8/2019 exam 3 lec m

50/58

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!!

8/8/2019 exam 3 lec m

51/58

8/8/2019 exam 3 lec m

52/58

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

!

!

8/8/2019 exam 3 lec m

53/58

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)

8/8/2019 exam 3 lec m

54/58

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

8/8/2019 exam 3 lec m

55/58

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

8/8/2019 exam 3 lec m

56/58

Bernoullis equation

(Derived from

conservation of energy)

tconsgyvP tan

2

1 2 ! VV

8/8/2019 exam 3 lec m

57/58

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

8/8/2019 exam 3 lec m

58/58

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