mechanics lecture 4, slide 1 comprehensive review a) exam information b) what kind of questions? c)...

Mechanics Lecture 4, Slide 1

Comprehensive Review

Comprehensive Reviewa) Exam informationb) What kind of questions?c) Review

Midterm 3 Exam in-class


Average = 5.1/12

Average = 57% (normalized to 9)

%100*943

correctedclassin NNScore

The score for Midterm 3 will be calculated with the following formula:

Final Exam Review the following material

homework problems. video pre-lectures/textbook. Lecture slides Unit Main Points

Multiple choice…but show your work and justification. Mostly Calculations…”step by step” Some Conceptual questions…like checkpoint problems. Bring calculators and up to ten sheets of notes.

It is best to prepare your own hand-written notes! Derived Equations may be helpful…(e.g. projectile

motion)


What we covered…


Kinematics Description of Motion

Force Dynamics-how objects change velocity

Energy Kinetic and Potential

Conservation Laws Momentum and Energy

Collisions Elastic and In-elastic

Rotations Torque/ Angular Momentum/Statics

Problem Solving Techniques


Visualize/Diagram “Sketch” problem Identify variables, input and what we are trying to solve Free-body diagrams

Express in Mathematical Equations Scalars-1d Vectors-2d,3d Break into components System of n-equations with n-unknowns Use Mathematical tools to solve:

Quadratic Equation Vector operations Trigonometry

Conceptual Understanding Does answer make sense?

Potential Problem Topics


Projectile MotionRelative Motion - 2dUniform Circular MotionForces

Weight (near earth) Gravitational (satellite) Springs Normal Force Tension Friction

Free-Body DiagramsWork-Kinetic EnergyPotential Energy

Center of MassConservation of

MomentumCollisions

In-elastic Elastic

Rotations Kinematics Dynamics Statics Moment of Inertia Torque

Angular Momentum

Relevant Formulae

Mechanics Review 2 , Slide 7

Relevant Formulae


Kinematics


HyperphysicsMotion


Displacement vs timet

Velocity vs timet

Acceleration vs timet

HyperphysicsMotion


1d-Kinematic Equations for constant acceleration


))((2)))((

2

1)(

)(

)(

020

2

002

0

00

0

xtxavtv

xtvtatx

vtatv

ata

Basic Equations to be used for 1d – kinematic problems.

Need to apply to each object separately sometimes with time offset

When acceleration changes from one constant value to another say a=0 The problem needs to be broken down into segments

Ballistic Projectile Motion Quantities


Initial velocityspeed,angle

Maximum Height of trajectory, h=ymax

Range of trajectory, D

Height of trajectory at arbitrary x,t

“Hang Time”Time of Flight, tf

Derived Projectile Trajectory Equations


2

000 cos2

1

cossin)(

v

xg

v

xvxy

200 2

1)( gttvyty y

Height of trajectory as f(x), y(x)

Height of trajectory as f(t) , y(t)

g

vD

2sin20

g

v

g

vt y

f

sin2200

g

vyh

2

sin 220

0

Range of trajectory

Time of Flight (“Hang Time”)

Maximum height

Relative Motion in 2 Dimensions


Speed relative to shore

Direction w.r.t shoreline

Uniform Circular Motion



Uniform Circular Motion

Constant speed in circular path

Acceleration directed toward center of circle

What is the magnitude of acceleration?

Proportional to:1. Speed

2. time rate of change of angle or angular velocity

fTdt

d 22

v = wR

R

vac

2

Dynamics


Inventory of Forces


Weight Normal Force Tension Gravitational Springs …Friction


http://hyperphysics.phy-astr.gsu.edu/hbase/N2st.html#c1

http://hyperphysics.phy-astr.gsu.edu/hbase/N2st.html#c1


m2

m2g

T

N

f

m1

m1g

T

m2

m1

g

1) FBD

1) FBD2) SF=ma

add

N = m2g

T – m m2g = m2a m1g – T = m1a

m1g – m m2g = m1a + m2a

a = m1g – m m2g

m1 + m2

m2

m2g

T

N

f

m1

m1g

T

m2

m1

g


T is smaller when a is bigger

m1g – T = m1a

T = m1g – m1aa = m1g – m m2g

m1 + m2

m2

m2g

T

N

f

m1

m1g

T

m2

m1

g


1) FBD2) SF=ma

Gravitation Problems…too!


2221

satellitemarscenter

satellitemars

r

mmG

r

mmGF


m

F

R

va

r

mmGF

gravc

grav

2

221

r

mGr

m

Fv

m

r

r

mmGr

m

Fv

rm

Frav

mars

sat

grav

sat

satmars

sat

grav

sat

gravc

2

Be careful what value you use for r !!! Should be distance between centers of mass of the two objects

Work-Kinetic Energy Theorem

The work done by force F as it acts on an object that moves between positions r1 and r2 is equal to the change in the object’s kinetic energy:


KW

ldFWr

r

2

1

2

2

1mvK

But again…!!!

Energy Conservation Problems in general


)()( tUtKUKUKE ffiimechanical

For systems with only conservative forces acting

0 mechanicalE

Emechanical is a constant

Determining Motion

Force Unbalanced Forces acceleration

(otherwise objects velocity is constant)

Determine Net Force acting on object

Use kinematic equations to determine resulting motion


Energy Total Energy Motion, Location

Work Conservative forces

Motion from Energy conservation

aF

m

Fa

1221 FF

UKEmechanical

veconservatinonmechanical WE

KldFWfr

r

netnet

0

0

KUE

KWU

mechanical

net

fmechanicaliif

fmechanicaliif

mechanicaliifinalmechanical

fffinalmechanical

UEUKm

v

UEUKK

EUKE

UKE

2

,

,

;...0vatv f

Friction


“It is what it has to be.”

Block


2

22

2sinsin

sin

2

2

1

t

xgmmamgf

m

fmg

m

Fa

t

xaatx

k

knet

Work & Kinetic Energy


Example Problem


xTldTWtension

20

2

20

2

2

12

1

vvmWW

vvmK

KWW

KWWW

ftensionfriction

f

tensionfriction

frictiontensionnet

Potential Energy


Example Problems


xhh

xhmgxmgmghmgh

WEE

xmgW

WE

ki

kiki

frictioninitialmechanicalfinalmechanical

kfriction

frictionmechanical

,,

Conserve Energy from initial to final position.

2v ghhh

Example: Pendulum


2

2

1mvmgh ghv 2

ghv 2

Gravitational Potential Problems


gravitymechanical hUhmvUKE )()(2

1 2

conservation of mechanical energy can be used to “easily” solve problems.

Define coordinates: where is U=0?

M

M

E

Etotal

M

MMMoon

E

EEEarth

rr

mGM

rr

mGMrU

r

mGMrU

r

mGMrU

)(

)(

)(

0)( r

mGMrU E as r

rEr

Err

Mr

Mrr

Add potential energy from each source.

Collisions


Center of Mass Conservation of

Momentum

Inelastic collisions

Elastic Collisions

Impulse and Reference Frames

Multiple particles, Solid Objects

00 tottot

ext Pdt

dPF

Isolated system, No external force

Non-conservative internal force

0

0

tot

tot

P

K

Conservative internal force

0

0

tot

tot

P

K

Individual Particle changes momentum due to Force acting over a given duration

Favg = DP/Dt

Systems of Particles


Example Problem


58.0231

125.0301

ii

iii

cm m

xmx

1

22

2

12

1

21

2

2

12

211

222

211

2

1

21

2

2

122

2

112

221121

2211

2121

00

m

m

mm

m

m

vmm

m

vm

vm

vm

K

K

vm

mv

m

vmv

vmvmmm

vmvm

m

vmv

ii

iii

cm

Collisions


Example Problem


fiif

ffii

ffftotal

iiitotal

total

vmvmvmm

v

vmvmvmvm

vmvmp

vmvmp

p

,11,22,112

,2

,22,11,22,11

,22,11,

,22,11,

1

0

222

2

12

,22,11,1,2

,22,112

,22

1

,21,22,112

,2

,1,2

3)(22

1)2(

2

1

0)

21

1(2

22

)1(

1)1(

1

mvvmvmK

m

mvvm

mm

m

vmvmvv

vmvmm

vm

m

vmvmvmm

v

vv

i

iiff

iif

fiif

ff

23

0

mvK

K f

Example Problem :


21

1,1,

*1,

*1

21

1,1,1,

*1

21

112211

21

1

1

1

mm

mvvv

mm

mvvvv

mm

vmvmvm

mmv

iif

iCMii

CM

Impulse


|Favg | = |DP | /Dt = 2mv cosq /Dt


Rotations


Rotational Kinematics

Moment of Inertia

Torque

Rotational Dynamics

Rotational Statics

Angular Momentum

Description of motion about a center of mass

Resistance to changes in angular velocity

Force applied at a lever arm resulting in angular acceleration

Newton’s 2nd law for rotations

How to ensure stability

Vector Quantity describing object(s) rotation about an axis

Rotational Kinematics


Rotational Dynamics


Example Problem


I

MRI

MLL

MMLI

MRI

hoop

endrod

disk

2

222,

2

3

1)

2(

12

12

1

Example Problem


Work & Energy (rotations)


Example Problem


f

f

f

FMa

R

a

RFMR

FMgMa

MgN

5

2

5

2

sin

cos

2

sin7

5

sin)5

21(

5

2sin

ga

ga

MaMgMa

sin7

2sin

7

5

5

25

2

MggMF

MaF

f

f

Statics


Statics Problems


Example Problem


Angular Momentum


Example Problem


Relevant Formulae


mechanics lecture 4, slide 1 comprehensive review a) exam information b) what kind of questions? c)...

Documents