kinetics: imp-mom (ch. 5 & 8) review -...
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Kinetics: Imp-Mom (Ch. 5 & 8) ReviewLecture 34
ME 231: Dynamics
Question of the Day
How do we go from F=ma to impulse and momentum?
Integrate equations of motion with respect to time
What is the relationship between impulse andWhat is the relationship between impulse and momentum?
Impulse (F*t or M*t) on a particle or body equals h i ( )change in momentum (G or H)
Why use impulse and momentum to solveWhy use impulse and momentum to solve dynamics problems?
Facilitates the solution of problems where forces acte ified ti e i t l d i t l ho t
2
over specified time interval or during extremely shortperiods of time (e.g., impact)
ME 231: Dynamics
Outline for Today
• Question of the day• Impulse-momentum principlesp p p• Linear momentum for rigid bodies• Angular momentum for rigid bodiesgu a o e tu o g d bod es• Equations, equations, equations…• Exam 2b breakdown (kinetics: imp-mom)Exam 2b breakdown (kinetics: imp mom)
• Exam 2a grades
3
• Exam 2a grades…
ME 231: Dynamics
Linear Momentum vG m�
• Particle of mass m islocated by positionvector r
• Velocity v is tangent t it thto its path
• The mass timesvelocity is thevelocity is the linear momentum
• Tangent to its path
4ME 231: Dynamics
Impulse-MomentumPrinciple: Linear vG m�
dtdtt
t
t
t
2
1
2
1�� �� GF �GF ��� � � � �x
t
t xx vmdtFvm 212
1
��� �� � � �t2
�21
2
1
GFG ��� � dtt
t
� � � �yt
t yy vmdtFvm 212
1
��� �� � � �t
vmdtFvm 2 ��� �� � � �zt zz vmdtFvm 211
��� �impulse-momentumdiagramdiagram
• Integrate to describe the effect of the resultantf li fi i
5
force �F on linear momentum over a finiteperiod of time
ME 231: Dynamics
Angular Momentum vrH mO ��O
• Particle of mass mis located by position vector rposition vector r
• Velocity v and linearmomentum G = mvmomentum G = mv are tangent to its path
• The moment of the linear momentum vectorThe moment of the linear momentum vectormv about point O is the angular momentumHO of P about O
6
• Perpendicular to plane A defined by r and vME 231: Dynamics
Impulse-MomentumPrinciple: Angular vrH mO �� OO HM ��Principle: Angular
• Integrate to describe th ff t f th l
dtdtt
O
t
O22
�� �� HM �
O OO
the effect of the angularimpulse �MO*t onangular momentum HO
t Ot O11��
� � � �212
O
t
OO dt HMH ��� � angular momentum HOof m about O over a finite period of time
� � � �211
Ot OO �
� � 21 )()( 2
1
zvyvmdtzvyvm yz
t
t xOyz ��� � M
� �2t
� � � 21 )()( 2
1
xvzvmdtxvzvm zx
t
t yOzx ��� � M
� � )()( 2 yvxvmdtyvxvmt
��� � M
7ME 231: Dynamics
� � 21 )()(1
yvxvmdtyvxvm xyt zOxy ��� � M
Impulse-Momentum: Exercise
A small car with mass of 20kg rolls freely and carries a 5 kg sphere mounted on a5-kg sphere mounted on a light rotating rod with r =0.4 m and angularvelocity of 4 rad/s. The car has a velocity v = 0.6 m/s when � 0ºm/s when � = 0º.
Determine v when � = 60º
8ME 231: Dynamics
Determine v when � = 60º.
Outline for Today
• Question of the day• Impulse-momentum principlesp p p• Linear momentum for rigid bodies• Angular momentum for rigid bodiesgu a o e tu o g d bod es• Equations, equations, equations…• Exam 2b breakdown (kinetics: imp-mom)Exam 2b breakdown (kinetics: imp mom)
• Exam 2a grades
9
• Exam 2a grades…
ME 231: Dynamics
Linear Momentum for Rigid Bodies
• Special case of a general system of particles
vG m� vG m
GF ��2 GFG ��� � dtt
�
GF � 211
GFG ��� � dtt
� � � �tGdFG 2 ��xx GF ��
yy GF ��
� � � �xt xx GdtFG 212
1
��� �� � � �t
GdtFG 212 ��� �
10ME 231: Dynamics
yy � � � �yt yy GdtFG 211
�� �
Angular Momentum for Rigid BodiesAbout the Mass Center G
angularlinear
G �H I
�
vG m�
�
�H GG I�
GG HM ��GF ��
212
1
GFG ��� � dtt
t� � � �21
2
1G
t
t GG dt HMH ��� �
11ME 231: Dynamics
Alternate Angular Momentum for Rigid BodiesAbout an Arbitrary Point P About a Fixed Point O
P
�
mvd�IH GP ��
�
�IH OO �
PP HM �
� � � �2t dt HMH ��� �
OO HM �
� � � �2t dt HMH ��� �12ME 231: Dynamics
� � � �211
Pt PP dt HMH ��� � � � � �211
Ot OO dt HMH ��� �
Impulse-Momentum: Exercise
Determine the time t requiredto bring the centrifuge to anangular velocity � fromangular velocity � fromrest under a constanttorque M applied tothe shaft.
13ME 231: Dynamics
Outline for Today
• Question of the day• Impulse-momentum principlesp p p• Linear momentum for rigid bodies• Angular momentum for rigid bodiesgu a o e tu o g d bod es• Equations, equations, equations…• Exam 2b breakdown (kinetics: imp-mom)Exam 2b breakdown (kinetics: imp mom)
• Exam 2a grades
14
• Exam 2a grades…
ME 231: Dynamics
Equations, Equations, Equations…Particle Kinetics: F=ma
Lecture Equations
Particle Kinetics: F=ma
18. Newton 2nd Law19. Eqs. of Motion20. Rectilinear
aF m�xmmaF xx ����
ymmaF yy ����zmmaF zz ����
21. Curvilinear rr maF �
�� maF �nn maF �
tt maF �t
27. Lin. Imp. Mom.GF ���vG m� 21
2
1
GFG ��� � dtt
t0G �
H � � � �2t d HMH ��28. Ang. Imp. Mom.vrH mO ��
OO HM ��� � � �21
2
1Ot OO dt HMH ��� �
0H � O
vG m� v�HH mGP ���� �iiiG m��H ���
15
29. Sys. Imp. Mom.
ME 231: Dynamics
vG m�� �iiiO mvrH ��
v�HH mGP ��� �iiiG m��H �
GG HM �� a�HM mGP ��� �
Equations, Equations, Equations…Rigid Body Kinetics: F=ma
Lecture Equations
Rigid Body Kinetics: F=ma
18. Newton 2nd Law22. Gen. Eqs. Mot. I23. Gen. Eqs. Mot. II
GG HM ��iiG F�H ���
mad�IM GP ��PPP mI a��M ���
F M I M Iq
24. Fixed-Axis Rot.
25. Gen. Plane Mot. I
2mrII GO ��aF m� �M GG I� �M OO I�
mad�IM GP ��PPP mI a��M ���
2OO mkI �
25. Gen. Plane Mot. I
26. Gen. Plane Mot. IIaF m� �M GG I�
GPPPP
vG m� �H GG I� mvd�IH GP �� �IH OO �
31. Body Imp. Mom. GF ��
�t
GG HM ��
� � � �t
�
PP HM �� OO HM ��
16ME 231: Dynamics
212
1
GFG ��� � dtt
t� � � �21
2
1G
t
t GG dt HMH ��� �
Exam 2a Breakdown (particle kinetics: F=ma)
40
25
30
35
oints
15
20
25
3440
3426
30
40
mber�of�po
5
10
5 26
num
018.�Newton�2nd�Law
19.�Eqs.�of�Motion
20.�Rectilinear 21.�Curvilinear
27.�Lin.�Imp.�Mom.
28.�Ang.�Imp.�Mom.
29.�Sys.� Imp.�Mom.
0
17ME 231: Dynamics
lecture
Exam 2 Breakdown (rigid body kinetics: F=ma)
70
50
60
points
30
40 70
40 40 40mber�of�p
10
20 34 40 4026
0
40
num
018.�Newton�2nd�Law
22.�Gen.�Eqs.�of�Motion�I
23.�Gen.�Eqs.�of�Motion�II
24.�Fixed�Axis�Rotation
25.�Gen.�Plane�Mot.� I
26.�Gen.�Plane�Mot.� II
31.�Body�Imp.�Mom.
0
18ME 231: Dynamics
lecture
Outline for Today
• Question of the day• Impulse-momentum principlesp p p• Linear momentum for rigid bodies• Angular momentum for rigid bodiesgu a o e tu o g d bod es• Equations, equations, equations…• Exam 2b breakdown (kinetics: imp-mom)Exam 2b breakdown (kinetics: imp mom)
• Exam 2a grades
19
• Exam 2a grades…
ME 231: Dynamics
Grades
95
105
85
65
75
grad
e
45
55
35
45
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61
20ME 231: Dynamics
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61
student
Homework Grades
95
105
85
65
75
grad
e
45
55
35
45
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61
21ME 231: Dynamics
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61
student
Homework and Exam 1 Grades
95
105
85
65
75
grad
e
45
55
35
45
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61
22ME 231: Dynamics
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61
student
Homework, Exam 1, and Exam 2a Grades
95
105
85
65
75
grad
e
45
55
35
45
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61
23ME 231: Dynamics
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61
student
Exam 1 Grades
25
20
udents
10
15
15
26
ber�of�stu
0
55 7
1215
numb
0
<60% 60�69% 70�79% 80�89% 90�100%
earned grade
24ME 231: Dynamics
earned�grade
Exam 2a Grades
25
20
udents
10
15
16 18
25
ber�of�stu
0
5
14n
umb
0
<60% 60�69% 70�79% 80�89% 90�100%
earned grade
25ME 231: Dynamics
earned�grade
Exam 2a Problem Grades
100
80
100
ge�(%)
40
60
70.0 76.0 88.0 85.0
s�averag
0
20
class
1 2 3 4
problem
26ME 231: Dynamics
problem
For Next Time…
• Review Chapters 5 & 8• Review Lectures slides
– http://rrg.utk.edu/resources/ME231/lectures.html
• Review Examples from class– http://rrg.utk.edu/resources/ME231/examples.html
Exam #2b on Friday (11/16)• Exam #2b on Friday (11/16)
27ME 231: Dynamics