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0593_C11_fm Page 391 Monday, May 6, 2002 2:59 PM

Generalized Dynamics: Kinematics and Kinetics 391

The force S exerted on Q by the spring is:

S =−F n =−kx n (11.11.8)

The generalized force F relative to x is then:
x
F =⋅Sv = − kx (11.11.9)
x x ˙
Let a potential energy function P be defined as:

P = ( )kx 2 (11.11.10)
12
Then, from Eq. (11.11.1), we have:

x
P
−∂ ∂ = −kx = F (11.11.11)
x
which is consistent with Eq. (11.11.9).
It happens that Eqs. (11.11.5) and (11.11.10) are potential energy functions for gravity
and spring forces in general. Consider first Eq. (11.11.5) for gravity forces. Suppose Q is
a particle with mass m. Let Q be a part of a mechanical system S having n degrees of
freedom represented by the coordinates q (r = 1,…, n). Recall from Eq. (11.6.5) that the
r
ˆ
contribution of the weight force on Q to the generalized force F for the coordinate q is:
F
r r r
ˆ
∂
F =− mg h q ∂ (11.11.12)
r r
where h is the elevation of Q above a reference level as in Figure 11.11.3.
From Eq. (11.11.2), if a potential energy function P is given by mgh, Eq. (11.11.1) gives
ˆ
the contribution to the generalized force of q for the weight force as:
F
r r
ˆ
∂
F =−∂ ∂P q =− mg h q ∂ (11.11.13)
r r r
which is consistent with Eq. (11.11.12).
Consider next Eq. (11.11.10) for spring forces. Suppose Q is a point at the end of a spring
which is part of a mechanical system S as depicted in Figure 11.11.4. Let S have n degrees

S S
Q(m)
k
O
n
h(q ,t)
r
Q

R R

FIGURE 11.11.3 FIGURE 11.11.4
Elevation of a particle Q of a mechanical system S. A spring within a mechanical system S.