
417Motion of Rigid Bodies
© 2010 Taylor & Francis Group, LLC
where
′
=
−
+V
Ia b
Mgh()
(cos )
sin
cosθ
θ
θ
2
2
2
(12.110)
can be considered as an “effective potential.” Equation 12.109 can be solved to yield t(θ):
t
EV I
()
)]
θ
θ
θ
=
′
−
′
∫
d
/2
(12.111)
This integral can be inverted to give θ(t). But detailed integration involves elliptic integrals.
Physics will then be obscured in the profusion of mathematics. Fortunately, some qualitative fea-
tures of the motion can be obtained by other means as outlined below.
If we plot V′(θ) against θ (Figure 12.25) in the physically limited range 0 ≤ θ < π, we have an
energy diagram of the kind discussed in Chapter 6 on cent ...