phy1020.DVI

8.1 Equation of Motion

Since the pendulum (in the small-angle approximation) is a simple harmonic oscillator, its motion is given by
(cf. Eq. (5.2))

.t/D 0 cos.!tCı/; (8.3)

where 0 is the (angular) amplitude in radians andıis the phase constant. To find the angular frequency!,
note from geometry that the horizontal displacement distance of the pendulum isxDLsin. Writing Eq.
(8.2) as

F

mg

L



.Lsin/; (8.4)

and comparing with Eq. (5.1), we can see that the effective spring constant for the pendulum is

keffD

mg

L

: (8.5)

Now for the harmonic oscillator we know!D

p

k=m, and so

!D

r

keff

m

D

r

mg

mL

(8.6)

or

!D

r

g

L

: (8.7)

So the small-amplitude motion of the simple plane pendulum is the same as the mass on a spring; but the
angular frequency of the spring system is given by!D

p

k=m, and for the pendulum it is!D

p
g=L.
Other simple harmonic oscillators with have other expressions for their angular frequency!, each depending
on the physical parameters of the system.

8.2 Period

Since the period of a simple harmonic oscillator is given byT D2=!, we find, using Eq. (8.7), that the
period of the pendulum is

TD2

s

L

g

: (8.8)

Remember that this is just anapproximateexpression for the period of a pendulum, with the approxima-
tion being better the smaller the amplitude 0. An exact treatment requires the periodTto be expressed as
an infinite series. The details require some advanced mathematics that is beyond the scope of this course, but
if you’re interested, an exact treatment of the simple plane pendulum is given in Appendix R.

8.3 The Spherical Pendulum

Aspherical pendulumis similar to a simple plane pendulum, except that the pendulum is not constrained to
move in a plane; the massmis free to move in two dimensions along the surface of a sphere. Figure 8.2
shows a photograph of the movement of a spherical pendulum.