Lecture
3
Vector Mechanics for Engineers:
Dynamics
MECN 3010
Department of Mechanical Engineering
Inter American University of Puerto Rico
Bayamon Campus
Dr. Omar E. Meza Castillo
omeza@bayamon.inter.edu
http://www.bc.inter.edu/facultad/omeza
Inter - Bayamon
Tentative Lecture Schedule
Topic
Lecture
Kinematics of a Particle
1
Kinetics of a Particle: Force and Acceleration
Kinetics of a Particle: Work and Energy
Kinetics of a Particle: Impulse and Momentum
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Planar Kinematics of a Rigid Body
2
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"Lo peor es educar por métodos basados
en el temor, la fuerza, la autoridad,
porque se destruye la sinceridad y la
confianza, y sólo se consigue una falsa
sumisión”
Einstein Albert
Topic 1: Kinematics of a
Particle
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Introduction and Basic Concepts
3
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Chapter Objectives
 To introduce the concepts of position,
displacement, velocity, and acceleration.
 To study particle motion along a straight
line and represent this motion graphically.
 To investigate particle motion along a
curve path using different coordinate
systems.
 To present an analysis of dependent
motion of two particles.
 To examine the principles of relative
motion of two particles using translating
axes.
4
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12.7 Curvilinear Motion: Normal and Tangential Components
When the path along which a particle travels is known, then it is
often convenient to describe the motion using n and t coordinate
axes which act normal and tangent to the path
a. Planar Motion: The t axis is
tangent to the curvature, and n
is perpendicular to the t axis.
ρ(rho) is the radius of curvature
and O’ is the center of
curvature.
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The plane which contains the n
and t axes is referred as the
embracing or osculating plane.
ds
Radius of curvature
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12.7 Curvilinear Motion: Normal and Tangential Components
b.
Velocity: Since the particle
moves, s is a function of time.
The particle’s velocity v has a
direction that is always tangent
to the path, and a magnitude
that is determined by taking the
time derivative of the path
function s=s(t), i.e., v= ds/dt.
Hence
v=vut
where
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v  s
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12.7 Curvilinear Motion: Normal and Tangential Components
c. Acceleration: The acceleration of
the particle is the time rate of
change of the velocity. Thus
where
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where
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12.8 Curvilinear Motion: Cylindrical Components
a. Polar Coordinates: If motion is
restricted to the plane, then
polar coordinates is used.
We can specify the location of
the particle shown in figure
using a radial coordinate r(ur),
which extend outward from the
fixed origin O to the particle,
and a transverse coordinate
θ(uθ),
which
is
the
counterclockwise angle between
a fixed reference line and the r
axis.
b. Position: At any instant the
position of the particle, is
defined by the position vector
r = rur
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12.8 Curvilinear Motion: Cylindrical Components
c.
Velocity: The instantaneous
velocity v is obtained by taking
the time derivative of r. Using a
dot to represent the time
derivative, we have
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where
where
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12.8 Curvilinear Motion: Cylindrical Components
d. Acceleration: Taking the time
derivatives of the last velocity
equations,
we
obtain
the
particle’s
instantaneous
acceleration
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where
where
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12.8 Curvilinear Motion: Cylindrical Components
e. Cylindrical Coordinates: IF the
particles moves along a space
curve as shown in figure, then
its location may be specified by
the three cylindrical coordinates
r,θ,z.
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The position, velocity, and
acceleration of the particle can
be written in terms of its
cylindrical
coordinates
as
follows:
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Homework2  WebPage
MECN 4600
Due, Tuesday, February 06, 2012
Omar E. Meza Castillo Ph.D.
27
Lecture 3