8.7 Physics Project: the pendulum 263
dθ
dtˆ=vˆ,
and
dvˆ
dtˆ
=−
vˆ
Q
−sin(θ)+Aˆcos(ωˆˆt).These are the equations to be solved. The factorQrepresents the number of oscillations
of the undriven system that must occur before its energy is significantly reduced due to the
viscous drag. The amplitudeAˆis measured in units of the maximum possible gravitational
torque whileωˆ is the angular frequency of the external torque measured in units of the
pendulum’s natural frequency.
8.7 Physics Project: the pendulum
8.7.1 Analytic results for the pendulum
Although the solution to the equations for the pendulum can only be obtained through nu-
merical efforts, it is always useful to check our numerical code against analytic solutions. For
small anglesθ, we havesin(θ)≈θand our equations become
dθ
dtˆ
=vˆ,and
dvˆ
dtˆ=−
vˆ
Q−θ+
Aˆcos(ωˆtˆ).These equations are linear in the angleθand are similar to those of the sliding block or the
RLC circuit. With given initial conditionsvˆ 0 andθ 0 they can be solved analytically to yield
θ(t) =[
θ 0 − Aˆ(^1 −ωˆ(^2) )
( 1 −ωˆ^2 )^2 +ωˆ^2 /Q^2
]
e−τ/^2 Qcos(√
1 − 4 Q^12 τ)+[
vˆ 0 + 2 θQ^0 −
Aˆ( 1 − 3 ωˆ^2 )/ 2 Q
( 1 −ωˆ^2 )^2 +ωˆ^2 /Q^2]
e−τ/^2 Qsin(√
1 − 4 Q^12 τ)+
Aˆ( 1 −ωˆ^2 )cos(ω τˆ )+ωQˆsin(ω τˆ)
( 1 −ωˆ^2 )^2 +ωˆ^2 /Q^2 ,and
vˆ(t) =[
vˆ 0 −
Aˆωˆ^2 /Q
( 1 −ωˆ^2 )^2 +ωˆ^2 /Q^2]
e−τ/^2 Qcos(√
1 − 41 Q 2 τ)−[
θ 0 + 2 vˆ^0 Q−
Aˆ[( 1 −ωˆ^2 )−ωˆ^2 /Q^2 ]
( 1 −ωˆ^2 )^2 +ωˆ^2 /Q^2]
e−τ/^2 Qsin(√
1 − 4 Q^12 τ)+
ωˆAˆ[−( 1 −ωˆ^2 )sin(ω τˆ)+Qωˆcos(ω τˆ)]
( 1 −ωˆ^2 )^2 +ωˆ^2 /Q^2 ,withQ> 1 / 2. The first two terms depend on the initial conditions and decay exponentially
in time. If we wait long enough for these terms to vanish, the solutions become independent
of the initial conditions and the motion of the pendulum settles down to the following simple
orbit in phase space
θ(t) =Aˆ( 1 −ωˆ^2 )cos(ω τˆ )+ωQˆsin(ω τˆ )
( 1 −ωˆ^2 )^2 +ωˆ^2 /Q^2 ,and
vˆ(t) =ωˆAˆ[−( 1 −ωˆ^2 )sin(ω τˆ )+Qωˆcos(ω τˆ )]
( 1 −ωˆ^2 )^2 +ωˆ^2 /Q^2 ,tracing the closed phase-space curve