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Particle Kinetics: Normal and Tangential Coordinates (Solving Problems…
Particle Kinetics: Normal and Tangential Coordinates
Normal and Tangential Coordinates
When a particle moves along a
curved path
, it may be more convenient to write the equation of motion in terms of
normal and tangential coordinates
The normal direction (n)
always
points towards the path's
center of curvature
. In a circle, the center of curvature is the center of the circle
The tangential direction (t) is
tangent
to the path,
usually set as positive in the direction of motion of the particle
Equations of Motion
Since the equation of motion is a
vector
equation, Σ
F
= m
a
, it may be written in terms of the n and t coordinates as ΣF(tangential)
u(tangential)
+ ΣF(normal)
u(normal)
+ ΣF(binormal)
u(binormal)
= m
a(tangential
+ m
a(normal)
Here ΣF(tangential) and ΣF(normal) are the sums of the force components acting in the t and n directions, respectively.
This vector equation will be satisfied provided the individual component on each side of the equation are equal, resulting in the two
scalar
equations: ΣF(tangential) = ma(tangential) and ΣF(normal) = ma(normal)
Since there is no motion in the binormal (b) direction, we can also write ΣF(binormal) = 0
Normal and Tangential Accelerations
The
tangential acceleration
, a(tangential) = dv/dt, represents the time rate of
change in the magnitude of the velocity
. Depending on the direction of ΣF(tangential), the particle's speed will either be increasing or decreasing.
The
normal acceleration
, a(normal) = v^2 / ρ, represents the time rate of
change in the direction
of the velocity vector. Remember, a(normal)
always
acts towards the path's center of curvature. Thus, ΣF(normal) will always be directed toward the center of the path
If the path of motion is defined as y = f(x), the
radius of curvature
at any point can be obtained from:
ρ = [1 + (dy/dx)^2]^(3/2) / |d2y/dx2|
Solving Problems With n-t Coordinates
Use n-t coordinates when a particle is moving along a
known, curved path
.
Place the
n-t coordinate system
on the particle
Draw
Free-Body-Diagrams
and
Kinetic Diagrams
of the particle. The
normal acceleration
(a(normal)) always acts "inwards" (the positive n-direction). The
tangential acceleration
(a(tangential)) may act in either the positive or negative t direction.
Apply the
equations of motion
in scalar form and solve. It may be necessary to employ the
kinematic relations
: a(tangential) = dv/dt = vdv/ds ; a(normal) = v^2 / ρ
