Foucault/Bravais pendulum applet
G.Barenboim and J.A.Oteo
is well known among scientists that Foucault pendulum
experiment is a demonstration of the Earth's rotation.
Less known, by far, is the
so-called Bravais pendulum which is characterized by
describing conical trajectories in space.
Whereas the oscillation
plane of the Foucault pendulum seems to rotate, in the
Bravais pendulum the two senses for the
conical rotation are not
equivalent, which constitutes also a proof of Earth's
Hence, Foucault and
Bravais pendula are two extreme cases of a general
oscillation mode defined by the initial conditions.
Foucault pendulum starts
its motion from a non-equilibrium point with vanishing
velocity. Bravais pendulum needs a particular
value of tangential
velocity in order to describe conical oscillations.
The system is the very same for both pendula.
In between, different
initial conditions give rise to beautiful patterns
created by the trajectory followed by the pendulum
This applet plots the trajectory described by the bob of a
in the rotating Earth with constant angular velocity at
Small amplitudes $x(t)$, $y(t)$ in two dimensions is the
approximation considered in solving the
motion equations [1,2].
To proceed one has first to write down the initial conditions for
$x_0, y_0, x'_0, y'_0$;
as well as the parameter
$L$: length of the pendulum.
$g=9.8 m/s$: gravity acceleration.
$\omega=\sqrt(g/L)$: angular frequency of the
$\Omega$: angular velocity of the reference
$t-final$: time when the trajectory plotting
Central panel buttons
provide default values that mimic some
especial cases, namely:
Foucault: FOUCAULT PENDULUM. Initial displacement
vanishing initial velocity.
Bravais: BRAVAIS PENDULUM or CONICAL
PENDULUM. Initial displacement $x_0$
and tangential initial velocity
The pendulum oscillates creating a conical shape.
General: GENERAL PENDULUM (or rather
whimsical) initial conditions.
General, but with $\Omega \sin(\beta)=0.000075 rad/s$,
namely the true Earth angular velocity and the
pendulum placed at the north pole ($\sin \beta =1$).
By default, $L=10 m$ so that $\omega=1$ rad/s and
$\Omega=0.1$ rad/s (except in the case Earth).
The trajectories in left and right panels differ only in
sign of the initial velocity. The right panel is the slave
trajectory provided the initial conditions
and parameter values
are properly written down.
REFRESH: Erases the trajectory plotted till
that instant and
plotting from there.
ALT+Left mouse button:
detaches the applet from the navigator.
 G.L.Baker and J.A.Blackburn, "The pendulum: a case
study in physics"
(Oxford University Press, 2005).
 G.Barenboim and J.A.Oteo, "The linear Foucault and
Bravais pendula revisited"
(work in preparation).