Page 343 - Dynamics of Mechanical Systems

0593_C10_fm Page 324 Monday, May 6, 2002 2:57 PM

324 Dynamics of Mechanical Systems

t *
∫ ⋅
W = Fv dt (10.2.13)
0
*
where t is the time of action of F. The integrand of Eq. (10.2.13), F • v, is often called the
power of F (see Section 10.4).
As a second example, consider the work done by gravity on a simple pendulum as it
falls from a horizontal position to the vertical equilibrium position as in Figure 10.2.4. Let
the pendulum mass be m and let its length be as shown. The gravity (or weight) force
is, then,

w = mg k (10.2.14)

where k is a vertically downward directed unit vector as shown in Figure 10.2.4.
To apply Eq. (10.2.2), consider that the differential arc vector may be expressed as:

ds = ldφ n φ (10.2.15)

where φ measures the angle of the pendulum to the horizontal and n is a unit vector
φ
tangent to the circular arc of the pendulum as shown in Figure 10.2.4. Hence, the work
of the weight force is:

= /
/
φπ 2 π 2
W = ∫ wds ∫ mg ⋅ n φ dφ
⋅
l
k
φ =0 0
(10.2.16)
π 2/ π 2
/
d =
= mgl ∫ cos φφ mg sin φ = mgl
l
0
0
Observe from the next to last expression of Eq. (10.2.16) that the work done by gravity
as the pendulum falls through an arbitrary angle φ is:
=
Wmgl sinθ (10.2.17)
The distance sinφ may be recognized as the vertical drop h of the pendulum; hence,
the gravitational work is:

W = mgh (10.2.18)

P(m)
k O
φ

n
φ
FIGURE 10.2.4
P(m)
A falling pendulum.