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Lissajous Figure |
The figures commonly called Lissajous Figures were investif=gated by Jules Antoine Lissajous in the 1850s, but had earlier been generated and studied by Nathaniel Bowditch of Salem, Massachusetts in 1815, who built a special pendulum to generate them. Later, a Scottish Professor named Hugh Blackborn would create a similar device, which would devlop into the "Harmonograph". The figures it generated would become a 19th century craze.
The basic Lissajous figure is made up of two sinusoidal oscillations along directions at right angles to each other, and when the periods of the osillations along both directions are in whole-number ratio to each other (such as 1:2, or 2:3. The abiove figure has a ratio of 3:4). The simplest way to achieve this is by using a pendulum whose support has a "Y" shape.
Solution
System:
Interactions:
Model:
Approach:
Diagrammatic Representation
First, consider the Y support for the pendulum:
The pendulum effectively has length L2 when swinging in the horizontal plane in and out of the page, but length L1 along the horizontal direction in the plane of the page.
Mathematical Representation
The force is supplied by a belt around the smaller wheel of radius r (in a 19th century factory, it would probably be a circular leather belt attached to the water wheels). This means that the direction the force is applied along is always tangential to the circumference of the wheel, and hence Torque = r X F = rF
\begin
[ \vec
= \vec
X \vec
= rF = I_
\alpha ]\end
The Moment of Inertia of combined bodies about the same axis is simply the sum of the individual Moments of Inertia:
\begin
[ I_
= I_
+ I_
]\end
The Moment of Inertia of a solid disc of radius r and mass m about an axis through the center and perpendicular to the plane of the disc is given by:
\begin
[ I = \frac
m r^2 ] \end
So the Moment of Inertia of the complete flywheel is:
\begin
[ I_
= \frac
(m r^2 + M R^2 ) ]\end
The expression for the angular velocity and the angle as a function of time (for constant angular acceleration) is given in the Laws of Change section on the Rotational Motion page:
\begin
[ \omega_
= \omega_
+ \alpha (t_
- t_
) ] \end
and
\begin
[ \theta_
= \theta_
+ \omega_
( t_
- t_
) + \frac
\alpha ( t_
- t_
)^2 ]\end
We assume that at the start, ti = 0 , we have both position and angular velocity equal to zero. The above expressions then simplify to:
\begin
[ \omega_
= \alpha t_
]\end
and
\begin
[ \theta_
= \frac
\alpha {t_{\rm f}}^2 ]\end
where
\begin
[ \alpha = \frac
{I_{\rm total}} = \frac
]\end