Page 341 - Dynamics of Mechanical Systems

0593_C10_fm Page 322 Monday, May 6, 2002 2:57 PM

322 Dynamics of Mechanical Systems

P

P d
F

P

F C

FIGURE 10.2.1 FIGURE 10.2.2.
A force F moving a particle P. A force F applied to a particle P moving on a curve C.

In view of these observations, it is necessary to generalize the definition of Eq. (10.2.1):
specifically, let a force F be applied to a particle P which moves along a curve C as in
Figure 10.2.2. Then, the work W done by F as P moves along C is defined as:

δ
⋅
W = ∫ F ds (10.2.2)
D
0
where ds is a differential arc length vector tangent to C at the position of P and where δ
is the distance P moves along C under the action of F.
From this generalized definition of work we see that if F is always directed perpendicular
to the movement of P, the work is zero. Also, we see that if P moves in a direction opposed
to F, the work is negative.
As an example, consider a particle P moving on a curve C defined by the parametric
equations:

,
r
x = cos t y = sin t z = t (10.2.3)
r
,
where C may be recognized as a circular helix as depicted in Figure 10.2.3. Let F be a force
defined as:
F = t n + t 2 n + t 3 n (10.2.4)
x y z

Z
n
z

P C
p
O
Y
n y
FIGURE 10.2.3 X n x
A particle moving on a circular helix.