Velocity and Acceleration in Polar Coordinates
In polar coordinates (r, θ)one can employ the orthogonal unit vectors
ˆer= cosθˆe 1 + sin θeˆ 2
ˆeθ=−sin θˆe 1 + cos θˆe 2
(9 .22)
to represent the position of a moving particle.
If in two-dimensional polar coordinates the position vector of a moving particle
is represented in the form
r =reˆr
where r,θ and consequently, eˆr, ˆeθ are changing with respect to time t, then the
velocity of the particle is given by
v =dr
dt
=rdˆer
dt
+dr
dt
ˆer (9 .23)
Differentiate the vectors in equation (9.22) with respect to time tand show
dˆer
dt
=−sin θdθ
dt
ˆe 1 + cos θdθ
dt
ˆe 2 =dθ
dt
ˆeθ
dˆeθ
dt
=−cos θdθ
dt
ˆe 1 −sin θdθ
dt
ˆe 2 =−dθ
dt
ˆer
(9 .24)
The first equation in (9.24) simplifies the equation (9.23) to the form
v =drdt =rdθdt ˆeθ+drdt ˆer= ̇rˆer+rθ ̇ˆeθ (9 .25)
where ̇ = dtd. Here vr= ̇ris called the radial component of the velocity and the term
vθ=rθ ̇is called the transverse component of the velocity or tangential component of
the velocity. The speed of the particle is given by
v=|v |=
√
(rθ ̇)^2 + ̇r^2
which represents the magnitude of the velocity.
The acceleration of the particle is obtained by differentiating the velocity. Dif-
ferentiate the equation (9.25) and show
a =
dv
dt =
d^2 r
dt^2 = ̇r
dˆer
dt + ̈r
ˆer+rθ ̇dˆeθ
dt +
d
dt(r
θ ̇)ˆeθ
= ̇r(θ ̇ˆeθ) + ̈rˆer+rθ ̇(−θ ̇ˆer) + (rθ ̈+ ̇rθ ̇)ˆeθ
a =( ̈r−r(θ ̇)^2 )ˆer+ (rθ ̈+ 2 ̇rθ ̇)ˆeθ
