Kinetics: F=ma (Ch. 3 & 7) Reviewrrg.utk.edu/resources/ME231/lectures/ME231_lecture_32.pdf ·...
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Kinetics: F=ma (Ch. 3 & 7) ReviewLecture 32
ME 231: Dynamics
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Question of the Day
What is the most important concept in mechanics?
ME 231: Dynamics
Free Body Diagram
What is the most important concept in dynamics?
Equations of Motion
aF m��xmmaF xx �����
ymmaF yy �����
zmmaF zz �����
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Outline for Today
� Question of the day� Where are we in the course?� Inverse vs. forward dynamics� Kinetics: cause of motion� Possible solutions to kinetics problems� Direct application of Newton’s 2nd Law� Plane motion types for rigid bodies� Equations, equations, equations…� Exam 2a breakdown (kinetics: F=ma)
ME 231: Dynamics
� Question of the day
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Kinematics Kinetics Dynamics
Where are we in the course?
ME 231: Dynamics
Chapters 1, 2, 6 Chapters 3, 5, 7, 8
Relationship among position, velocity, and acceleration
Relationship among forces(and moments)andacceleration
Concept: What is dynamics?
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F = m a
Where are we in the course?
ME 231: Dynamics
Acceleration. Velocity rate of change with
respect to time
Calculation: How do we use dynamics?
Newton’s 2nd Law
Force. A push or pull exerted on a body, characterized by:�magnitude�direction�point of application
Mass. Measure of the resistance of a body to linear acceleration.
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Inverse vs. Forward Dynamics
ME 231: Dynamics
aF m�inverse
dtd
dtd
Positions
Velocities
Forces
aF m�
ME 231: Dyna
forward
� � Positions
Velocities
Forces
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Statics Kinematics Kinetics
Kinetics: Cause of Motion?
ME 231: Dynamics
Chapters 1, 2, 6 Chapters 3, 5, 7, 8
Relationship among position, velocity, and acceleration
Relationship among forces(and moments)andacceleration
Concept: What is kinetics?
ME 202
Relationship among forces (and moments)and equilibrium
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Possible Solutions to Kinetics Problems
� Direct application of Newton’s 2nd Law – force-mass-acceleration method– Chapters 3 and 7
� Use of impulse and momentum methods– Chapters 5 and 8
� Use of work and energy principles – Chapter 4
ME 231: Dynamics
� Use of impulse and momentum methods– Chapters 5 and 8
� Use of work and energy principles– Chapter 4
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Step-by-Step Solution Process
1. Kinematics– Identify type of motion– Solve for linear and angular accelerations
2. Diagram– Assign inertial coordinate system– Draw complete free-body diagram– Draw kinetic diagram to clarify equations
3. Equations of motion– Apply 2 linear and 1 angular equations– Maintain consistent sense– Solve for no more than 5 scalar unknowns (3 scalar
equations of motion and 2 scalar relations from the relative-acceleration equation)
ME 231: Dynamics
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Outline for Today
� Question of the day� Where are we in the course?� Inverse vs. forward dynamics� Kinetics: cause of motion� Possible solutions to kinetics problems� Direct application of Newton’s 2nd Law� Plane motion types for rigid bodies� Equations, equations, equations…� Exam 2a breakdown (kinetics: F=ma)
ME 231: Dynamics
� Question of the day� Where are we in the course?� Inverse vs. forward dynamics� Kinetics: cause of motion� Possible solutions to kinetics problems
![Page 11: Kinetics: F=ma (Ch. 3 & 7) Reviewrrg.utk.edu/resources/ME231/lectures/ME231_lecture_32.pdf · Possible solutions to kinetics problems Direct application of Newton’s 2 nd Law Plane](https://reader033.fdocuments.net/reader033/viewer/2022041413/5e19b0f47253e2253015f38b/html5/thumbnails/11.jpg)
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Direct Application of Newton’s 2nd Law
ME 231: Dynamics
rF ��m�� aF m��or
iim rfF �������iim rfffFFF ���� �������� 321321
xx amF ��
yy amF ��
zz amF ��
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Rectangular (x-y) Coordinates: Exercise
ME 231: Dynamics
A particle with mass of 10 slugs moving in two-dimensions has a position vector (r) as a function of time (t) with coordinates given by
where r is measured in feet and t is in seconds.
Determine the magnitude of the net force (F) accelerating the particle at time t = 3 s.
x(t) = t2 – 4t + 20 , y(t) = 3 sin(2t)
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Polar (r-��) Coordinates: Exercise
Tube A rotates about the vertical O-axis with constant angular velocity � and contains a small cylinder B of mass m whose radial position is controlled by a cord passing through the tube and wound around a drum of radius b.
ME 231: Dynamics
� � ���� eea r 2 2 ������� rrrr ���
Determine the tension T in the cord and ��-component of force F� if the drum has a constant angular rate of rotation of �� as shown.
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Normal and Tangential (n-t) Coordinates: Exercise
A 1500-kg car enters an s-curve and slows down from 100 km/h at A to a speed of 50 km/h as it passes C.
Determine the total horizontal force exerted by the road on the tires at positions A, B, and C.
ME 231: Dynamics
tn eea 2
vv���
�
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Outline for Today
� Question of the day� Where are we in the course?� Inverse vs. forward dynamics� Kinetics: cause of motion� Possible solutions to kinetics problems� Direct application of Newton’s 2nd Law� Plane motion types for rigid bodies� Equations, equations, equations…� Exam 2a breakdown (kinetics: F=ma)
ME 231: Dynamics
� Question of the day� Where are we in the course?� Inverse vs. forward dynamics� Kinetics: cause of motion� Possible solutions to kinetics problems� Direct application of Newton’s 2nd Law
![Page 16: Kinetics: F=ma (Ch. 3 & 7) Reviewrrg.utk.edu/resources/ME231/lectures/ME231_lecture_32.pdf · Possible solutions to kinetics problems Direct application of Newton’s 2 nd Law Plane](https://reader033.fdocuments.net/reader033/viewer/2022041413/5e19b0f47253e2253015f38b/html5/thumbnails/16.jpg)
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B
A
B
A
B
A
B
A
B’
A’
B’
A’
B’
A’
B
APlane Motion Types for Rigid Bodies
� Translation
� Fixed-axis rotation
� General plane motion
ME 231: Dynamics
BBBBBBBBBB
A
B’BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
A’AAAA
B’
A’
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17ME 231: Dynamics
Rigid-Body Translation
curvilinearrectilinear
aF m��
0
0
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���
A
P
GG
MmadM�IM
00
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Rigid-Body Translation: Exercise
A cleated conveyor belt transports solid cylinders up a 15º incline. The diameter of each cylinder is half its height.
Determine the maximum acceleration for the belt without tipping the cylinders as it starts.
ME 231: Dynamics
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Fixed-Axis Rotation
ME 231: Dynamics
free-bodydiagram
kineticdiagram
� Mass center’s circular motion easily expressed in n-t coordinates
� Plane-motion equations: aF m�� �M GG I�� �M OO I��
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Fixed-Axis Rotation: Exercise
Determine the angular acceleration and the force on the bearing at O for (a) the narrow ring of mass m and (b) the flat circular disk of mass m immediately after each is released from rest with OC horizontal.
ME 231: Dynamics
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General Plane Motoin:Combined Translation and Rotation
ME 231: Dynamics
aF m�� �M GG I��
mad�IM GP ���
PPP mI a��M ����
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General Plane Motion: Exercise
A truck has a mass of 2030 kg and carries a 1500-mm-diameter spool of cable with a massof 0.75 kg per meter of length. There are 150 turns on the full spool. The empty spool has a mass of 140 kg with radius of gyration of 530 mm.
Determine the tension Tin the cable when the truck starts from rest with an acceleration of 0.2g.
ME 231: Dynamics
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Outline for Today
� Question of the day� Where are we in the course?� Inverse vs. forward dynamics� Kinetics: cause of motion� Possible solutions to kinetics problems� Direct application of Newton’s 2nd Law� Plane motion types for rigid bodies� Equations, equations, equations…� Exam 2a breakdown (kinetics: F=ma)
ME 231: Dynamics
� Question of the day� Where are we in the course?� Inverse vs. forward dynamics� Kinetics: cause of motion� Possible solutions to kinetics problems� Direct application of Newton’s 2nd Law� Plane motion types for rigid bodies
![Page 24: Kinetics: F=ma (Ch. 3 & 7) Reviewrrg.utk.edu/resources/ME231/lectures/ME231_lecture_32.pdf · Possible solutions to kinetics problems Direct application of Newton’s 2 nd Law Plane](https://reader033.fdocuments.net/reader033/viewer/2022041413/5e19b0f47253e2253015f38b/html5/thumbnails/24.jpg)
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Lecture Equations
18. Newton 2nd Law19. Eqs. of Motion20. Rectilinear
21. Curvilinear
27. Lin. Imp. Mom.
28. Ang. Imp. Mom.
29. Sys. Imp. Mom.
Equations, Equations, Equations…Particle Kinetics: F=ma
ME 231: Dynamics
aF m��xmmaF xx �����
ymmaF yy �����zmmaF zz �����
rr maF ��
�� maF ��nn maF ��
tt maF ��
GF ���vG m� 21 2
1
GFG ��� � dtt
t0G ��
vrH mO ��
OO HM ���� � 21 2
1O
t
t OO dt HMH ��� �0H �� O
vG m�� iiiO m vrH ���
v�HH mGP ���� iiiG m ��H ����
GG HM ��� a�HM mGP ���� �
27. Lin. Imp. Mom.
28. Ang. Imp. Mom.
29. Sys. Imp. Mom.
GF ���vG m� 21
2
1
GFG ��� �1
dtt
��t��0G ��
vrH mO ��
OO HM ���� � 21
2
1O
t
t OO dt HMH ��� �1
tt
tt0H �� O
vG m�� iiiO m vriH ���
v�HH mGP ���� iiiG m ��H ����
GG HM ��� a�HM mGP ���� �
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Lecture Equations
18. Newton 2nd Law22. Gen. Eqs. Mot. I23. Gen. Eqs. Mot. II
24. Fixed-Axis Rot.
25. Gen. Plane Mot. I
26. Gen. Plane Mot. II
31. Body Imp. Mom.
Equations, Equations, Equations…Rigid Body Kinetics: F=ma
ME 231: Dynamics
GG HM ���iiG F�H ����
mad�IM GP ���PPP mI a��M ����
2mrII GO ��aF m�� �M GG I�� �M OO I��
aF m�� �M GG I��mad�IM GP ���PPP mI a��M ����
vG m�
GF ���
21 2
1
GFG ��� � dtt
t
GG HM ���
� � 21 2
1G
t
t GG dt HMH ��� �
�H GG I� mvd�IH GP ��
PP HM ��� OO HM ���
�IH OO �
2OO mkI �
31. Body Imp. Mom.
vG m�
GF ���
212
1
GFG ��� �1
dtt
��t��
GG HM ���
� � 212
1G
t
t GG dt HMH ��� �1
tt
tt
�H GG I� mvd�IH GP ��
PP HM ��� OO HM ���
�IH OO �
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Exam 2a Breakdown (particle kinetics: F=ma)
ME 231: Dynamics
0
5
10
15
20
25
30
35
40
18. Newton 2nd Law
19. Eqs. of Motion
20. Rectilinear 21. Curvilinear
27. Lin. Imp. Mom.
28. Ang. Imp. Mom.
29. Sys. Imp. Mom.
3440
0
3426
30
40
nu
mb
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27. Lin. Imp. Mom.
28. Ang. Imp.Mom.
29. Sys. Imp.
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Mom.
40
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Exam 2 Breakdown (rigid body kinetics: F=ma)
ME 231: Dynamics
0
10
20
30
40
50
60
70
18. 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.
34
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0
40
nu
mb
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31. Body Imp. Mom.
40
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For Next Time…
� Review Chapters 3 & 7� Review Lectures slides
– http://rrg.utk.edu/resources/ME231/lectures.html
� Review Examples from class– http://rrg.utk.edu/resources/ME231/examples.html
� Exam #2a on Friday (11/9)
ME 231: Dynamics