Page 371 - Advanced engineering mathematics

11.2 Velocity and Curvature 351

function of t, not of s. We, therefore, usually compute the curvature as a function of t by using
the chain rule:

dT dt
dt ds
κ(t) =
1

= T (t) .

F (t)
EXAMPLE 11.3
Let C have position vector
2
F(t) =[cos(t) + t sin(t)]i +[sin(t) − t cos(t)]j + t k.
for t ≥ 0. Figure 11.5 is part of the graph of C. A tangent vector is given by
F (t) = t cos(t)i + t sin(t)j + 2tk.

This tangent vector has the length
√

v(t) = F (t) = 5t.
The unit tangent vector in terms of t is
1 1

T(t) = F (t) = √ [cos(t)i + sin(t)j + 2k].

F (t) 5
Then
1

T (t) = √ [−sin(t)i + cos(t)j],
5
and the curvature of C is
1

κ(t) = T (t)

F (t)

1 1 2 1
2
= √ [sin (t) + cos (t)]=
5t 5 5t

80
–6
60 –4
–4 40
–2 –2
20
0 0
2

2 4
6
4
8
6

8
FIGURE 11.5 Graph of the curve of Example 11.3.

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October 15, 2010 16:11 THM/NEIL Page-351 27410_11_ch11_p343-366