Environmental Engineering Reference
In-Depth Information
1.3.4 Velocity and acceleration in plane-polar co-ordinates: uniform
circular motion
Circular motion arises frequently in physics. Examples may be as simple as a
mass whirled on a string, but also include the orbits of satellites around the Earth,
and the motion of charged particles in a magnetic field. Where circular motion
is concerned, problems are often most easily solved in polar co-ordinates. In
this section we determine the equations for the velocity and acceleration in polar
co-ordinates.
The position of a particle moving in a plane is
r
=
r
r
.
(1.21)
As the particle moves, both
r
and
r
may change, i.e. they are both implicitly
time-dependent. The velocity of the particle is calculated by differentiation of the
product
r
r
.
d
d
t
(r
r
)
d
r
d
t
r
r
d
r
v
=
=
+
d
t
.
(1.22)
Since
r
=
cos
θ
i
+
sin
θ
j
,
d
r
d
t
sin
θ
d
θ
cos
θ
d
θ
d
θ
d
t
ˆ
=−
+
=
d
t
i
d
t
j
θ
,
(1.23)
ˆ
where we have used the definition Eq. (1.8) for
θ
. Thus,
d
r
d
t
r
r
d
θ
d
t
ˆ
v
=
+
.
(1.24)
θ
ˆ
The tangential contribution
r
d
d
t
is zero if the particle moves radially (constant
θ
)
θ
d
r
d
t
r
is zero for motion in a circle (constant
r
). We intro-
whereas the radial velocity
duce the angular speed
ω
=
d
θ/
d
t
, to simplify the notation. The velocity is then
d
r
d
t
r
rω
ˆ
v
=
+
.
(1.25)
θ
The general expression for acceleration can be obtained by differentiation of
Eq. (1.24) and further application of Eq. (1.8). However, at this point we will
concern ourselves with the case of uniform circular motion, i.e.
d
r
d
t
0and
ω
constant. In which case, we only need worry about the tangential term in (1.24) and
=
d
d
r
r
d
ω
d
t
d
t
(rω
ˆ
d
t
ω
ˆ
ˆ
rω
2
r
,
a
=
)
=
θ
+
θ
−
(1.26)
θ
wherewehaveused
d
ˆ
d
t
=−
ω
r
.
