phy1020.DVI

R.2 Solution,.t/ ........................................

If the amplitude 0 is small, we can approximate sin, and find the position.t/at any timetis given
by Eq. (8.3) in Chapter 8. But when the amplitude is not necessarily small, the anglefrom the vertical at
any timetis found (by solving Eq. (R.5)) to be a more complicated function:

.t/D 2 sin^1

ksn

r

g

L

.tt 0 /Ik



; (R.6)

where sn.xIk/is aJacobian elliptic functionwith moduluskDsin. 0 =2/. The timet 0 is a time at which
the pendulum is vertical (D 0 ) and moving in theCdirection.
The Jacobian elliptic function is one of a number of so-called “special functions” that often appear in
mathematical physics. In this case, the function sn.xIk/is defined as a kind of inverse of an integral. Given
the function

u.yIk/D

Zy

0

dt

p

.1t^2 /.1k^2 t^2 /

; (R.7)

the Jacobian elliptic function is defined as:

sn.uIk/Dy: (R.8)

Values of sn.xIk/may be found in tables of functions or computed by specialized mathematical software
libraries.

R.3 Period

As found in Chapter 8, the approximate period of a pendulum for small amplitudes is given by

T 0 D2

s

L

g

: (R.9)

This equation is really only anapproximateexpression for the period of a simple plane pendulum; the smaller
the amplitude of the motion, the better the approximation. Anexactexpression for the period is given by

TD 4

s

L

g

Z 1

0

dt

p

.1t^2 /.1k^2 t^2 /

; (R.10)

which is a type of integral known as acomplete elliptic integral of the first kind.
The integral in Eq. (R.10) cannot be evaluated in closed form, but itcanbe expanded into an infinite
series. The result is

TD2

s

L

g

(

1 C

X^1

nD 1



.2n1/ŠŠ

.2n/ŠŠ

 2

sin2n



 0

2

)

(R.11)

D2

s

L

g

(

1 C

X^1

nD 1



.2n/Š

2 2n.nŠ/^2

 2

sin2n



 0

2

)

: (R.12)