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VECTOR MECHANICS FOR ENGINEERS:
DYNAMICS
Eighth Edition
Ferdinand P. Beer
E. Russell Johnston, Jr.
Lecture Notes:
J. Walt Oler
Texas Tech University
CHAPTER
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
11 Kinematics of Particles
425203
Engineering Dynamics
Lecture Note
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Contents
Introduction
Rectilinear Motion: Position,
Velocity & Acceleration
Determination of the Motion of a
Particle
Uniform Rectilinear-Motion
Uniformly Accelerated Rectilinear-
Motion
Motion of Several Particles: Relative
Motion
Motion of Several Particles:
Dependent Motion
Graphical Solution of Rectilinear-Motion
Problems
Curvilinear Motion: Position, Velocity &
Acceleration
Derivatives of Vector Functions
Rectangular Components of Velocity and
Acceleration
Motion Relative to a Frame in Translation
Tangential and Normal Components
Radial and Transverse Components
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Introduction
• Dynamics includes:
- Kinematics: study of the geometry of motion. Kinematics is used to
relate displacement, velocity, acceleration, and time without reference to
the cause of motion.
- Kinetics: study of the relations existing between the forces acting on a
body, the mass of the body, and the motion of the body. Kinetics is used
to predict the motion caused by given forces or to determine the forces
required to produce a given motion.
• Rectilinear motion: position, velocity, and acceleration of a particle as it
moves along a straight line.
• Curvilinear motion: position, velocity, and acceleration of a particle as it
moves along a curved line in two or three dimensions.
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Rectilinear Motion: Position, Velocity & Acceleration
• Particle moving along a straight line is said
to be in rectilinear motion.
• Position coordinate of a particle is defined by
positive or negative distance of particle from
a fixed origin on the line.
• The motion of a particle is known if the
position coordinate for particle is known for
every value of time t. Motion of the particle
may be expressed in the form of a function,
e.g., 326 ttx
or in the form of a graph x vs. t.
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Rectilinear Motion: Position, Velocity & Acceleration
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Rectilinear Motion: Position, Velocity & Acceleration
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Rectilinear Motion: Position, Velocity & Acceleration
• Consider particle with motion given by
326 ttx
• at t = 0,
• at t = 2 s,
• at t = 4 s,
• at t = 6 s,
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Determination of the Motion of a Particle
• Recall, motion of a particle is known if position is known for all time t.
• Typically, conditions of motion are specified by the type of acceleration
experienced by the particle. Determination of velocity and position requires
two successive integrations.
• Three classes of motion may be defined for:
- acceleration given as a function of time, a = f(t)
- acceleration given as a function of position, a = f(x)
- acceleration given as a function of velocity, a = f(v)
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Determination of the Motion of a Particle
• Acceleration given as a function of time, a = f(t):
tttx
x
tttv
v
dttvxtxdttvdxdttvdxtvdt
dx
dttfvtvdttfdvdttfdvtfadt
dv
00
0
00
0
0
0
• Acceleration given as a function of position, a = f(x):
x
x
x
x
xv
v
dxxfvxvdxxfdvvdxxfdvv
xfdx
dvva
dt
dva
v
dxdt
dt
dxv
000
202
12
21
or or
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Determination of the Motion of a Particle
• Acceleration given as a function of velocity, a = f(v):
tv
v
tv
v
tx
x
tv
v
ttv
v
vf
dvvxtx
vf
dvvdx
vf
dvvdxvfa
dx
dvv
tvf
dv
dtvf
dvdt
vf
dvvfa
dt
dv
0
00
0
0
0
0
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Sample Problem 11.2
Determine:
• velocity and elevation above ground at
time t,
• highest elevation reached by ball and
corresponding time, and
• time when ball will hit the ground and
corresponding velocity.
Ball tossed with 10 m/s vertical velocity
from window 20 m above ground.
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Sample Problem 11.2
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Uniform Rectilinear Motion
For particle in uniform rectilinear motion, the acceleration is zero and
the velocity is constant.
vtxx
vtxx
dtvdx
vdt
dx
tx
x
0
0
00
constant
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Uniformly Accelerated Rectilinear Motion
For particle in uniformly accelerated rectilinear motion, the acceleration of
the particle is constant.
atvv
atvvdtadvadt
dv tv
v
0
000
constant
221
00
221
000
00
0
attvxx
attvxxdtatvdxatvdt
dx tx
x
020
2
020
221
2
constant
00
xxavv
xxavvdxadvvadx
dvv
x
x
v
v
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Example: Kinematics relations
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Exercise
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Homework
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Motion of Several Particles: Relative Motion
• For particles moving along the same line, time
should be recorded from the same starting
instant and displacements should be measured
from the same origin in the same direction.
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Sample Problem 11.4
Ball thrown vertically from 12 m level
in elevator shaft with initial velocity of
18 m/s. At same instant, open-platform
elevator passes 5 m level moving
upward at 2 m/s.
Determine (a) when and where ball hits
elevator and (b) relative velocity of ball
and elevator at contact.
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Sample Problem 11.4
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Sample Problem 11.4
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Motion of Several Particles: Dependent Motion
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Sample Problem 11.5
Pulley D is attached to a collar which
is pulled down at 3 cm/s. At t = 0,
collar A starts moving down from K
with constant acceleration and zero
initial velocity. Knowing that velocity
of collar A is 12 cm/s as it passes L,
determine the change in elevation,
velocity, and acceleration of block B
when block A is at L.
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Sample Problem 11.5
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Sample Problem 11.5
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Exercise
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Homework