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2
Question of the Day
What is the most important concept in Chapter 2?
ME 231: Dynamics
Time Derivative of a Vector
3
Outline for Today
� Question of the day� Where are we in the course?� Categories of motion� Choice of coordinate systems� Inertial versus moving coordinate systems� Degrees of freedom� Velocity and acceleration� Exam 1 breakdown (kinematics of
particles)ME 231: Dynamics
� Question of the day
4
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 acceleration
Concept: What is dynamics?
5
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.
6
Outline for Today
� Question of the day� Where are we in the course?� Categories of motion� Choice of coordinate systems� Inertial versus moving coordinate systems� Degrees of freedom� Velocity and acceleration� Exam 1 breakdown (kinematics of
particles)ME 231: Dynamics
� Question of the day� Where are we in the course?
7
Geometry of a problem identifies the category.
� Lecture 2: rectilinear (1D)
� Lecture 6: space (3D)
Geometry of a problemidentifies the category.
� Lecture 2: rectilinear (1D)
� Lecture 6: space (3D)
Categories of Motion
ME 231: Dynamics
Many of our motion problems involve curvilinear (2D), or plane motion.
� Lecture 3: curvilinear (2D)
�
8
Curvilinear (2D) Time Derivative: Exercise
ME 231: Dynamics
A particle moving in two-dimensions has a position vector (r) as a function of time (t) with coordinates given by
where r is measured in inches and t is in seconds.
Determine the velocity (v) and the acceleration (a).
x(t) = t2 – 4t + 20 , y(t) = 3 sin(2t)
9
Outline for Today
� Question of the day� Where are we in the course?� Categories of motion� Choice of coordinate systems� Inertial versus moving coordinate systems� Degrees of freedom� Velocity and acceleration� Exam 1 breakdown (kinematics of
particles)ME 231: Dynamics
� Question of the day� Where are we in the course?� Categories of motion
10
Motion of a problem identifies the coordinate system.Motion of a problem identifiestifies the coordinate system.
� Rectangular (x, y, z)� Polar (r, �, z)� Spherical (R, �, �)� Normal and
Tangential (n, t)
Choice of Coordinate Systems
ME 231: Dynamics
Many of our motion problems involve 2D rectangular coordinates (x, y).
� Polar (r, �, z)� Spherical (R, �, �)��� Normal and
Tangential (n, t)
11
Inertial Versus Moving Coordinate Systems
ME 231: Dynamics
Moving coordinate systems are measured with respect to an inertial coordinate system whose motion is negligible.
12
Inertial Versus Moving Coordinate Systems
ME 231: Dynamics
� Absolute position of B is defined in an inertial coordinate system X-Y
� Attach a set of translating (non-rotating) axes x-y to particle B and define the position of A
� Define position of “A relative to B” (“A/B”) in x-y
BABA /rrr ��
BABAA /rrrv ��� ���
BABAAA /rrrva ������� ����
13
Vector Representation: Exercise
ME 231: Dynamics
Train A travels with constant speed vA= 120 km/h. Anticipating the need to stop, car B decreases its speed of 90 km/h at the rate of 3 m/s2.
Determine the velocity and acceleration of the train relative to the car.
BABA /rrr ��
BABAA /rrrv ��� ���
BABAAA /rrrva ������� ����
14
Outline for Today
� Question of the day� Where are we in the course?� Categories of motion� Choice of coordinate systems� Inertial versus moving coordinate systems� Degrees of freedom� Velocity and acceleration� Exam 1 breakdown (kinematics of
particles)ME 231: Dynamics
� Question of the day� Where are we in the course?� Categories of motion� Choice of coordinate systems� Inertial versus moving coordinate systems
15
Degrees of Freedom
� Simple system of two interconnected particles
� With L, r2, r1, and b are constant
� Horizontal motion (x) of A is twice the vertical motion (y) of B
� Only one variable (x or y) is needed to specify the positions of all parts of the system
ME 231: Dynamics
yx �� 20 ��yx ���� 20 ��
BA vv 20 ��
BA aa 20 ��
bryrxL ����� 12 22
��Constraint Equations
16
Degrees of Freedom
ME 231: Dynamics
constant2 ��� DAA yyLConstraint Equations
� � constant���� DCCBB yyyyL
Position of lower cylinder depends on two variables (yA and yB)
DA yy �� 20 ��
DA yy ���� 20 �� CBA yyy ��� 420 ���
CBA yyy ������ 420 ���
DCB yyy ��� �� 20DCB yyy ������ �� 20
17
Degrees of Freedom: Exercise
ME 231: Dynamics
How many degrees of freedom are necessary to specify the position of all parts of the system of interconnected particles?
17111111111111111111111111111111111111111111111111111111111
18
Outline for Today
� Question of the day� Where are we in the course?� Categories of motion� Choice of coordinate systems� Inertial versus moving coordinate systems� Degrees of freedom� Velocity and acceleration� Exam 1 breakdown (kinematics of
particles)ME 231: Dynamics
� Question of the day� Where are we in the course?� Categories of motion� Choice of coordinate systems� Inertial versus moving coordinate systems� Degrees of freedom
19
Lecture Velocity Acceleration
2. Rectilinear
3. Curvilinear
4. Normal & Tangential
5. Polar Coordinates
7. Relative Motion
Velocity and Acceleration
ME 231: Dynamics
BABAA /rrrv ��� ��� BABAAA /rrrva ������� ����
�� eev r �� rr �� � � � � ���� eea r 2 2 ������� rrrr ���
tn eea 2
vv���
v � v et
jirv yx ��� ��� jirva yx ������� ����
2
2
0lim
dtsd
dtdv
tva
t��
��
� �dt
dstsv
t�
��
� � 0
lim
20
Outline for Today
� Question of the day� Where are we in the course?� Categories of motion� Choice of coordinate systems� Inertial versus moving coordinate systems� Degrees of freedom� Velocity and acceleration� Exam 1 breakdown (kinematics of
particles)ME 231: Dynamics
� Question of the day� Where are we in the course?� Categories of motion� Choice of coordinate systems� Inertial versus moving coordinate systems� Degrees of freedom� Velocity and acceleration
21
Exam 1 Breakdown (kinematics of particles)
ME 231: Dynamics
0
20
40
60
80
100
2. Rectilinear Motion
3. Curvilinear Motion
4. Normal & Tangential
5. Polar Coordinates
6. Space Motion
7. Relative Motoin
8. Constrained
Motion
6
96
10 10 0
83
4
nu
mb
er
of
po
ints
lecture
22
“Final” Course Grades (thru HW #5 LAST YEAR)
ME 231: Dynamics
0
5
10
15
20
25
<60% 60-69% 70-79% 80-89% 90-100% >100%
3 4
139
2219
nu
mb
er
of
stu
de
nts
calculated grade
23
“Final” Course Grades (thru HW #5 THIS YEAR)
ME 231: Dynamics
05
1015202530354045
<60% 60-69% 70-79% 80-89% 90-100% >100%
72 0
7
45
7
nu
mb
er
of
stu
de
nts
calculated grade