8.6 Physics examples 261
I=
dQ
dt,
we arrive at Eq. (8.18).
This section was meant to give you a feeling of the wide range of applicability of the
methods we have discussed. However, before leaving this topic entirely, we’ll dwelve into the
problems of the pendulum, from almost harmonic oscillations to chaotic motion!
8.6.3 The pendulum, a nonlinear differential equation
Consider a pendulum with massmat the end of a rigid rod of lengthlattached to say a fixed
frictionless pivot which allows the pendulum to move freelyunder gravity in the vertical plane
as illustrated in Fig. 8.5.
mg
mass m
length l
pivot
θ
Fig. 8.5A simple pendulum.
The angular equation of motion of the pendulum is again givenby Newton’s equation, but
now as a nonlinear differential equation
mld(^2) θ
dt^2
+mgsin(θ) = 0 ,
with an angular velocity and acceleration given by
v=l
dθ
dt
,
and
a=l
d^2 θ
dt^2.
For small angles, we can use the approximation