Acceleration
Outline of Hass, Thomas, Weir –
Section 10.5
The TNB Frame
Suppose that rt is a smooth curve that describes the
motion of a particle.
particle
The unit binormal vector, Bt, is defined to be
Bt  Tt  Nt.
Nt
The three vectors T, N, B form what is called the
TNB frame or the Frenet frame for the moving particle.
particle
These three vectors are mutually orthogonal to each other.
Tg
gives the forward direction of the motion.
N gives the direction in which the motion turns in the
plane of motion.
B gives
i
th direction
the
di ti in
i which
hi h the
th particle
ti l tends
t d to
t
twist out of the plane of motion.
Example
A particle travels along a helix with motion described by
x  a cost
y  a sint

z  bt.
Find Tt
Tt,, Nt
Nt,, and Bt
Bt..
Tangential and Normal Components of
Acceleration
l
The acceleration vector can be written as
at  a T tTt  a N tNt,
where
a T t

d 2s
dt 2
d
dt

|  |
|vt|
and
a N t  t
ds
dt
2
 t|vt|2 .
(Thi will
(This
ill be
b proved
d in
i class.)
l
)
a T t is called the tangential
g
component of acceleration.
a N t is called the normal component of acceleration.
In order to make writing things easier, let us agree to denote
speed
d by
b vt.
Th vt
thi notation,
t ti
we see that
th t
|  | With this
  Thus
   |vt|.
the tangential and normal components of acceleration are given
by
a T t  dv
dt
a N t
   tvt
    2 .
Remark
Recall that the velocity, vt, describes the speed and direction
off motion.
ti
A l ti
Accleration,
at,
i the
th rate
t off change
h
off velocity.
l it
  is
From the equation
at
   a T tTt
 T   a N tNt,
 N 
we see that a T t measures the rate of change
g of speed
p
and
that a N t measure the rate of change of direction.
Theorem
The normal component of acceleration, a N t, can be also
be found (without first finding the curvatuve) by using the formula
a N t 
|at|2  a T t 2 .
(This will be proved in class.)
Example
A particle travels along the parabola y  ax 2 with position
function given by the parametric equations
xt
y  at 2 .
Find at, a T t and a N t. Then write at in the form
at  a T tTt  a N tNt.