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336 Dynamics of Mechanical Systems

where the negative sign occurs because the upward movement of the pendulum is
opposite to the direction of gravity, producing negative work. The work–energy principle,
then, is:

W = ∆ K = K − K (10.8.15)
0 f f 0

or

(
2
−mgl 1 cosθ f ) ( ) ( ) ( ) ( ) 2 (10.8.16)
m lθ
˙
˙
m lθ
−
− 1 2
= 1 2
0
f
or
(
θ = θ − 2 g 1− cosθ f) (10.8.17)
˙ 2
˙ 2
l
f 0
If the pendulum is to rise all the way to the vertical equilibrium position (θ = π), we have:
θ = θ − 4 gl (10.8.18)
˙ 2
˙ 2
f 0
If θ ˙ 2 0 is exactly 4g , the pendulum will rise to the vertical equilibrium position and come
to rest at that position. If θ ˙ 2 exceeds 4g , the pendulum will rise to the vertical position
0
and rotate through it with an angular speed given by Eq. (10.8.18) (sometimes called the
rotating pendulum).

10.9 Elementary Example — A Mass–Spring System
For a third fundamental example, consider the mass–spring system depicted in Figure
10.9.1. It consists of a block B with mass m and a linear spring, with modulus k, moving
without friction or damping in a horizontal direction. Let x measure the displacement of
B away from equilibrium.
Suppose B is displaced to the right (positive x displacement with the spring in tension)
a distance δ away from equilibrium. Let B be released from rest in this position. Questions
arising then are what is the speed v of B as it returns to the equilibrium position (x = 0),
and how far to the left of the equilibrium position does B go?

B
k
m

FIGURE 10.9.1 x
Ideal mass–spring system.