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Example 7.9
Find the tangent, normal, and curvature of the trajectory of a particle moving
in uniform circular motion of radius a and with angular frequency ω.

Solution: The parametric equation of motion is

r
Rt( ) = acos(ω t ê) + asin(ω t ê)
1 2
The velocity vector is
r
dR t() =− a sin(ω ω t ê) + a cos(ω ω t ê)
dt 1 2

and its magnitude is aω.
The tangent vector is therefore:

T ˆ ( )t =− sin( tê +ω ) cos( têω )
1 2
The normal vector is

N ˆ ( )t =− cos( tê −ω ) sin( têω )
1 2
The radius of curvature is

dT ˆ ()t
dt − ω cos( tê − ω sin( tê 1
ω )
ω )
κ()t = r = 1 2 = = constant
ω )ê +
ω )ê
dR () t − a ω sin( t 1 a ω cos( t 2 a
dt

Homework Problems
Pb. 7.23 Show that in 2-D the radius of curvature can be written as:

′′′ − ′′′
xy yx
κ=
2 3
(( ′ x ) 2 + y( ′) ) /2
where the prime refers to the first derivative with respect to time, and the
double-prime refers to the second derivative with respect to time.

© 2001 by CRC Press LLC