Chapt. 4
Two and ThreeDimensional Kinematics
45
a) The displacement vector
vector
r
in polar coordinates is
vector
r
=
r
ˆ
r,
where
r
is the magnitude of
vector
r
and ˆ
r
is unit vector pointing in the direction of
vector
r
. Differentiate
vector
r
with respect to time
t
to find
vectorv
using the product rule, but don’t for the moment try to
determine what
d
ˆ
r/dt
is.
b) There is a nonrigorous, but valid and convincing way of evaluating
d
ˆ
r/dt
. Draw a diagram
with ˆ
r
and ˆ
r
+ Δˆ
r
with angle Δ
θ
between them.
The Δˆ
r
and Δ
θ
are the changes in,
respectively, radial unit vector ˆ
r
and angular of position
θ
of the object in time Δ
t
. With
Δ
θ
in radians show that
Δˆ
r
≈
Δ
θ
ˆ
θ,
where recall
ˆ
θ
is always
π/
2 = 90
◦
clockwise from ˆ
r
. Now show
d
ˆ
r
dt
=
ω
ˆ
θ,
where
ω
=
dθ/dt
is the angular velocity. The angular velocity is usually in units of radians
per unit time.
c) Now write
vectorv
substituting for
d
ˆ
r/dt
.
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 Fall '06
 Buchler
 Physics, Acceleration, Circular Motion, Angular velocity, Velocity

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