Kinematics
Curvilinear Motion
Johannes Kepler
Overview
• Curvilinear Motion: Normal and Tangential
Components
• Curvilinear Motion: Cylindrical Coordinates
Curvilinear Motion: Normal and
Tangential Components
Introduction
• For a particle moving along a known
curvilinear path
• If we take the position at an instant as the
origin
• We can define coordinate axes tangential, and
normal to the path
Normal and Tangential Components
• Consider the particle
at s from a fixed point O
• t-axis is tangent to the
curve. n-axes is normal
to the curve
O
• ut and un are the
respective unit vectors
and are perpendicular to
each other
O’
n
s
un
ut
Position
t
Normal and Tangential Components
• We may view the path as consisting
of a series of infinitesimally small
arcs of length ds, radius of
O’
O’
curvature ρ, with
ds
center O’
ρ
ρ
ρ
ρ
• The n-axis is positive
ρ ρ
O
O’
towards the center of
ds
curvature.
ds
u
• The plane containing the n
Radius of curvature
and t axes is called the
embracing or osculating plane
t
Normal and Tangential Components
Velocity
• Direction: always tangential
to the path
• Magnitude: Since s = s(time)
ds so
v vut
v
O’
ρ
dt
ρ
v
where
v
s
Velocity
Normal and Tangential Components
Acceleration
• As can be seen velocity is continually changing
as the particle traverses the curved path
• Acceleration is the rate of change
velocity with respect to time
a
v
dv
dt
a vut
O’
d (vut )
dt
vu t
dθ
ρ
ρ
un
ds
u’t
ut
Normal and Tangential Components
• Lets us redraw the velocity unit
vectors at the infinitesimal scale
u't
ut
dut
dut
ut
d
un
dθ
u’t
ut
dut
and for small angles
or dut d ut
but note that dut is in the direction of a unit
vector in the normal direction, so we can replace
the unit vector, thus dut d un
• And the derivative with respect to time becomes
u t
u
n
Normal and Tangential Components
• From the properties of an arc we know that
ds = ρ dθ
(go back two slides)
so likewise, with respect to time,
s

so
u t
u
s
n
un
v
un
Normal and Tangential Components
• We can consider acceleration as having two
components, one tangential and the other
normal
a
at ut
anun
• Comparing with original equation (Slide 7)
O’
or
at v
at ds vdv
and
an
v
2
an
a
P
at
Acceleration
Normal and Tangential Components
• The magnitude of the acceleration
is therefore
O’
a
2
t
a
2
n
a
an
a
P
at
Acceleration
Conclusion
• Examples
Curvilinear Motion: Cylindrical
Coordinates
• In some cases the motion of a particle is
constrained on a path amenable to analysis
using cylindrical coordinates
• If the motion is restricted to a plane, then we
can use polar coordinates
Abraham de Moivre
Polar Coordinates
• Consider a system where we locate a particle
by a radial coordinate r, which extends from
an origin O, and an angle θ (also called
transverse coordinate) in radians
measured counterclockwise
θ
form a fixed reference line to
u
the axis of r.
u
r
• uθ and ur are unit vectors in
θ
the directions of increasing O
θ and r respectively
Position
r
θ
r
Position & Velocity
• At any instant, we can define the position
vector of the particle as
r
rur
θ
• The instantaneous velocity is
the time derivative of r
v r
•
u r ?
rur
ru r
uθ
r
θ
O
Position
r
ur
Velocity
• A change in r over a time interval ∆t will not
result in a change of the unit vector ∆r
• Rather a change ∆θ over an interval ∆t will
cause a change ∆r
u'r
ur
ur
• For small angles
and
ur
u
ur
Velocity
u r
ur
t
lim
t
0
lim
t
t
0
u
u
u r
• So we can rewrite instantaneous velocity in
component form as
v
where
vr ur
vu
v
vθ
vr
v
r
r
r
O
θ
Velocity
vr
Velocity
• The speed (or magnitude of velocity) is
therefore given by
v
r
2
2

r
Acceleration
• Differentiating the instantaneous velocity
equation yields the instantaneous acceleration
r  u r  u
• We must now determine u
a
v
rur
r u r
r  u
• Consider a small change ∆θ over an interval ∆t
u'
u
for small angles
u
u
u
u
but ∆uθ acts in the negative ur
direction, so we can change the
unit vector for its direction
Acceleration
u
•
thus
u
ur
u
t
lim
t
u
0
u
lim
t
0
t
ur
r
So we can rewrite instantaneous
velocity in component form as
a
where
ar
a
ar ur
r r  2
r  2r 
a
aθ
au
r
O
θ
Acceleration
ar
Acceleration
• The term


is called the angular acceleration
d2
dt 2
d d
dt dt
• The magnitude of the acceleration
a
r r  2
2
2



r
2r
• The acceleration is generally not
tangential to the path.
Cylindrical Coordinates
• If the particle moves along a 3-dimensional
curve (also called a space curve) then we can
apply cylindrical coordinates to analyze the
motion
Cylindrical Coordinates
• We can extend the concept of
polar coordinates to 3-dimnesions
to create cylindrical coordinates
r, θ, and z
rP
v
a
rur
zu z
rur
r u
(r r  2 ) ur
uz
rP
z
O
zu z
(r  2r ) u
θ
zu z
r
Questions & Comments
• Examples