Kinematics of Particles (Ch. 2) ReviewLecture 16

ME 231: Dynamics

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?

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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.

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)

�

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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

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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)

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Inertial Versus Moving Coordinate Systems

ME 231: Dynamics

Moving coordinate systems are measured with respect to an inertial coordinate system whose motion is negligible.

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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 ������� ����

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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 ������� ����

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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

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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

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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

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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

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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

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96

10 10 0

83

4

nu

mb

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of

po

ints

lecture

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“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

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“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

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For Next Time…

� Complete Homework #6 due on Tuesday (10/2)

� Review Chapter 6

ME 231: Dynamics