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14:440:222 Dynamics
(Lecture Note #2)
Instructor: Prof. Peng Song
Rutgers University
Today’s Lecture
• Curvilinear motion
• Rectangular coordinates
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View Full Document Applications
The path of motion of a plane
can be tracked with radar and
its x, y, and z coordinates
(relative to a point on earth)
recorded as a function of time.
How can we determine the
velocity or acceleration of the
plane at any instant?
Applications (cont’d)
A roller coaster car travels
down a fixed, helical path at a
constant speed.
How can we determine its
position or acceleration at
any instant?
If you are designing the track, why is it important
to be able to predict the acceleration of the car?
General Curvilinear Motion
A particle moving along a curved path undergoes curvilinear motion.
Since the motion is often threedimensional, vectors are used to describe
the motion.
The position of the particle at any instant is designated by the vector
r
=
r
(t).
Both the magnitude and direction of
r
may vary with time.
A particle moves along a curve defined
by the path function, s.
If the particle moves a distance Ds along the
curve during time interval
±
t,
the
displacement is determined by vector
subtraction:
±²
r
=
r’

r
Velocity
Velocity represents the rate of change in the position of a
particle.
The average velocity of the particle
during the time increment
±
t
is
v
avg
=
±
r
/
±
t.
The instantaneous velocity is the
timederivative of position
v
= d
r
/dt .
The velocity vector,
v
, is always
tangent to the path of motion.
The magnitude of
v
is called the speed.
Since the arc length Ds
approaches the magnitude of
±
r
as
t±0,
the speed can be
obtained by differentiating the path function
(v = ds/dt).
Note
that this is not a vector!
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View Full Document Acceleration
Acceleration represents the rate of change in the
velocity of a particle.
If a particle’s velocity changes from
v
to
v’
over a
time increment
±
t,
the average acceleration during
that increment is:
a
avg
=
±
v
/
±
t=
±(
v

v’
)/
±
t
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This document was uploaded on 10/31/2011 for the course ENG MECH 232 at Rutgers.
 Fall '11
 R.C.Hibbeler

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