Curvilinear Motion: Normal and
Tangential Components
If you recall when curvilinear motion of a particle is studied in an x, y and z
rectangular coordinate system, its position is represent by position vector r
Instantaneous velocity,
Instantaneous acceleration,
In general r, v and a are all three dimensional Cartesian vectors.
Don’t forget
this
important conclusion
that the velocity of
the particle at any
point
is
always
tangent to the path.
Now lets look at this 3D curve path.
It can be divided into small
segments of curves with equal
lengths.
When the segment gets small enough,
each one of them approaches an arc,
which is a segment of a circle. And we
know that a circle always fall in a 2D
plane
For the next small segment of the path
it can also be approximated by another
arc that belong to another circle .
And then for another segment of the
path again it can be approximated by
an arc that belong to a circle.
Each segment ds is formed from the arc
of an associated circle having a radius of
curvature ρ and center of curvature O'.
For the particle travelling this arc
location, we can define a pair of axes
from it.
The first one is the t-axis being tangent
to the arc and other one is the n-axis
pointing toward the center of curvature.
It is also normal to the arc.
And with the definition of the t
tangent axis and n normal axis we can
represent the motion vectors using
the
tangential
and
normal
components instead of the x,y and z
rectangular components.
So for a particle in a short moment dt, if it travels along this curve path from location P to
Pʹ.
The distance travelled is the length of the arc ds on this path. At any given time, we can
always set up a pair of axes from the particle.
The t axis is tangent to the curve at the point and is positive in the direction of increasing
s. We will designate this positive direction with the unit vector Ut .
The normal axis n is perpendicular to the t axis with its positive sense directed toward the
center of curvature O and will be designated by the unit vector Un.
Velocity:
Since the particle moves and its position changes, s is a function of time.
The particle's velocity v has a direction that is always tangent to the
path, Fig. 12-24c, and a magnitude that is determined by taking the
time derivative of the path function ,
i.e., v = ds/ dt
ds 
v
s
V  vut
dt
ds
Acceleration:
The acceleration of the particle is the time rate of
change of the velocity.Thus,


dV d
a
 (vut )  vu t  vu (1)
dt dt
u’t = ut + dut
V  vut
ds 
v
s
dt
Two Special Cases of acceleration
at
Polar Coordinates
Curvilinear Motion: Polar Coordinates
e
Transverse: Situated or lying across; crosswise.
er
Curvilinear Motion: Polar Coordinates
Example:
Cardioid: A heart-shaped plane curve, the locus of a fixed point on a circle that rolls on the circumference
of another circle with the same radius.
r  0.5(1 cos )
ft
v4
s
ft
a  30 2
s
  180

?

?
Cos180  1
Sin180  0
Example:

  4rad / s
When   45,
Velocity  ?
Acceleration  ?
cos 
Base
100

Hypotenuse
r
r
90
 100m
Problem Sheet
RC Hibbler
Normal-Tangential Coordinate System
12-106; 12-117; 12-118; 12-119; 12-124
Radial-Transverse Coordinate System
12-145; 12-150; 12-152; 12-154; 12-164
Thank You