Foucault/Bravais pendulum applet

Conceived by G.Barenboim and J.A.Oteo

Developed by A.Fernández-García

This applet plots the trajectory described by the bob of a pendulum

in the rotating Earth with constant angular velocity at latitude $\beta$.

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 the bob:

$x_0, y_0, x'_0, y'_0$;

as well as the parameter values:

$L$: length of the pendulum.

$g=9.8 m/s$: gravity acceleration.

$\omega=\sqrt(g/L)$: angular frequency of the pendulum.

$\Omega$: angular velocity of the reference system.

$t-final$: time when the trajectory plotting stops.

Central panel buttons provide default values that mimic some

especial cases, namely:

Foucault: FOUCAULT PENDULUM. Initial displacement $x_0$ with

vanishing initial velocity.

Bravais: BRAVAIS PENDULUM or CONICAL PENDULUM. Initial displacement $x_0$

and tangential initial velocity $y'_0=x_0(\omega-\Omega)$.

The pendulum oscillates creating a conical shape.

General: GENERAL PENDULUM (or rather whimsical) initial conditions.

Earth: Like 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).

Plotting panels:

The trajectories in left and right panels differ only in the

sign of the initial velocity. The right panel is the slave one.

Control buttons:

PLAY: Plots trajectory provided the initial conditions

and parameter values are properly written down.

STOP: Stops plotting.

REFRESH: Erases the trajectory plotted till that instant and

then continues plotting from there.

ALT+Left mouse button: detaches the applet from the navigator.

References:

[1] G.L.Baker and J.A.Blackburn, "The pendulum: a case study in physics"

(Oxford University Press, 2005).

[2] G.Barenboim and J.A.Oteo, "The linear Foucault and Bravais pendula revisited"

(work in preparation).

Conceived by G.Barenboim and J.A.Oteo

Developed by A.Fernández-García

It
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 rotation.

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 bob.

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 rotation.

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 bob.

This applet plots the trajectory described by the bob of a pendulum

in the rotating Earth with constant angular velocity at latitude $\beta$.

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 the bob:

$x_0, y_0, x'_0, y'_0$;

as well as the parameter values:

$L$: length of the pendulum.

$g=9.8 m/s$: gravity acceleration.

$\omega=\sqrt(g/L)$: angular frequency of the pendulum.

$\Omega$: angular velocity of the reference system.

$t-final$: time when the trajectory plotting stops.

Central panel buttons provide default values that mimic some

especial cases, namely:

Foucault: FOUCAULT PENDULUM. Initial displacement $x_0$ with

vanishing initial velocity.

Bravais: BRAVAIS PENDULUM or CONICAL PENDULUM. Initial displacement $x_0$

and tangential initial velocity $y'_0=x_0(\omega-\Omega)$.

The pendulum oscillates creating a conical shape.

General: GENERAL PENDULUM (or rather whimsical) initial conditions.

Earth: Like 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).

Plotting panels:

The trajectories in left and right panels differ only in the

sign of the initial velocity. The right panel is the slave one.

Control buttons:

PLAY: Plots trajectory provided the initial conditions

and parameter values are properly written down.

STOP: Stops plotting.

REFRESH: Erases the trajectory plotted till that instant and

then continues plotting from there.

ALT+Left mouse button: detaches the applet from the navigator.

References:

[1] G.L.Baker and J.A.Blackburn, "The pendulum: a case study in physics"

(Oxford University Press, 2005).

[2] G.Barenboim and J.A.Oteo, "The linear Foucault and Bravais pendula revisited"

(work in preparation).