Spherical Pendulum

length of mass

1

mass

1

θ(0)

1.57

∂

t

θ(0)

0.

ϕ(0)

0.

angular momentum

2

solution time

30

The top of a pendulum of length

l

hangs from the origin. The mass

m

at the bottom end of the pendulum has coordinates

xlsin(θ)cos(φ)

,

ylsin(θ)sin(φ)

,

z-lcos(θ)

, where the vector



r

(t)

from the origin to

m

is at an angle θ to the negative

z

axis. The spherical coordinates of

m

are (

l

,

θ

,

φ

) with

∂

t

l0

. The Lagrange function and equations give

θ(t)

,

φ(t)

, and



r

(t)

. The integration constants are

θ(0)

,

φ(0)

,

∂

t

θ(0)

, and the angular momentum

p

φ

. The movement of the spherical pendulum is constrained to the spherical shell between

θ

min

and

θ

max

for all

φ

values. The pendulum cannot reach the singular points

θ0

and

θπ

for

p

φ

≠0

. When the angular momentum vanishes, the pendulum moves in a plane.

External Links

Foucault Pendulum (ScienceWorld)

Permanent Citation

Franz Krafft

"Spherical Pendulum"
http://demonstrations.wolfram.com/SphericalPendulum/
Wolfram Demonstrations Project
Published: January 15, 2008