Computational Physics - Department of Physics

262 8 Differential equations

sin(θ)≈θ.

and rewrite the above differential equation as

d^2 θ

dt^2

=−g

l

θ,

which is exactly of the same form as Eq. (8.15). We can thus check our solutions for small
values ofθagainst an analytical solution. The period is now

T=

2 π

√

l/g

.

We do however expect that the motion will gradually come to anend due a viscous drag
torque acting on the pendulum. In the presence of the drag, the above equation becomes

ml

d^2 θ

dt^2

+ν

dθ

dt

+mgsin(θ) = 0 ,

whereνis now a positive constant parameterizing the viscosity of the medium in question.
In order to maintain the motion against viscosity, it is necessary to add some external driving
force. We choose here, in analogy with the discussion about the electric circuit, a periodic
driving force. The last equation becomes then

mld

(^2) θ
dt^2
+νdθ
dt
+mgsin(θ) =Acos(ωt), (8.19)
withAandωtwo constants representing the amplitude and the angular frequency respec-
tively. The latter is called the driving frequency.
If we now define the natural frequency
ω 0 =

√

g/l,

the so-called natural frequency and the new dimensionless quantities

tˆ=ω 0 t,

with the dimensionless driving frequency

ωˆ=ω

ω 0

,

and introducing the quantityQ, called thequality factor,

Q=

mg

ω 0 ν

,

and the dimensionless amplitude

Aˆ= A

mg

we can rewrite Eq. (8.19) as

d^2 θ

dtˆ^2

+

1

Q

dθ

dˆt

+sin(θ) =Aˆcos(ωˆtˆ).

This equation can in turn be recast in terms of two coupled first-order differential equations
as follows