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Elliptical movement of a svinging pendulum

  1. Jun 24, 2009 #1
    As a school project I was assigned to measure the swing of a pendulum. At the time I was doing the experiment I found it almost impossible to prevent the pendulum from moving in an ellipse, at first i describe the elliptical movement as a source of error, but then it occurred to me that it might not matter for the reason that I was only measuring the movement from a 2D dimensional perspective from which the movement perpendicular to the ideal swing has no effect on the swing in the measured axis..

    What I am asking is, if an elliptical movement of a swing, is the same as a strait line movement in a 2D point of view?

    Am I right?
    Last edited: Jun 24, 2009
  2. jcsd
  3. Jun 24, 2009 #2


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    You can think of your viewing of the pendulum as an orthographic projection onto a viewing plane. If you allow the viewing plane to be described by two vectors, one of which is fixed in such a way as to point in the same vertical direction as your pendulum (set by gravity), then any rotation about this vector of the viewing plane will cause a distortion of the projected path only along the direction of the second vector. In short, the vertical positions of the pendulum will remain the same, but the horizontal positions that are projected will be distorted. In this sense it is easy to see that the projection can be of an ellipse. Think of the pendulum's circular arc and now rotate this arc in 3D space about the axis of gravity. The start and stop points of the arc in the vertical coordinate system remain constant but they will come closer in the horizontal direction, essentially squeezing the arc into an elliptical path.

    A better way to do the experiment would be to use a rigid pendulum or a bicycle chain which would to allow you to fix the swing of the pendulum to a desired plane. A way to fix the plot of the elliptical path would be to track the pendulum in a second plane that is perpendicular to the plane you made your projection from. This way, you will track the horizontal position (that is the position perpendicular to the axis defined by gravity), as a function of time, by mapping it to two vectors lying in the horizontal plane that are mutually normal. This would allow you to plot the pendulum's path in 3D coordinate space and make the appropriate rotation and translation to view the pendulum's path as lying only in a single plane which should result in a circular arc.
  4. Jun 24, 2009 #3
    thank you for your reply, but don't you mean the opposite? that the vertical position will change as a result of a perpendicular swing, and that the horizontal position will remain unchanged?

    I'm assuming that the horizontal axis is parallel to the line of the pendulum and that the vertical axis is in the direction of the primary swing,

    The purpose of the experiment is to measure the time of an amount of periods, the angle between the line of the pendulum and the direction of gravity, and the damping of the system

    will an elliptical swing affect this?
  5. Jun 24, 2009 #4


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    This is really nothing more than a simple drafting problem.

    Take a look at the attached JPEG where I have tried to illustrate my previous post. When you did your measurements, if the pendulum was swinging in a plane that is not parallel to the picture plane (the plane that projected against to track its motion), then it will be marked out as an ellipse. On the left is the desired picture. At the top left is a horizontal line representing the top view looking down on the swinging pendulum which marks out a straight line. If we project this down to the picture plane (the long horizontal line) and then draw out the front view, the pendulum will move in a perfect circular arc. Now let us rotate the pendulum's plane of oscillation to be at an angle with the picture plane, which is drawn on the right. At the top right you see that the pendulum is now swinging at an angle but it still marks out a path of the same length. If we project this down to the picture plane and draw the front view, we have three points of reference for the vertical positions. We know the two points at the ends of the pendulum path are the apex of the swing and the middle point is the lowest point of the swing. So we project these over and find the intersections. The dased line shows a circular arc made from the apex of the swing but the actual path is shown by the solid line, which is elliptical.

    Thus, if the pendulum is swinging at an angle to how you are projecting it when you map out the path, then it will appear elliptical. You can correct this by having a second map that is normal to the picture plane (a side view or the top view like I have shown) and using these two views you can reproduce the desired projection on the left in a process that is basically the opposite of what I explained above.

    As for your question about the angle, it is obvious from our image that the since the angle is dependent upon the horizontal position of the pendulum's apex, then the angle will be incorrect. It will be smaller than the actual angle.

    EDIT: I should make the caveat that the actual elliptical path shown in the picture is probably not the actual one since it was produced by only the projection of three points. Really what you would do is you would project a large number of points, maybe 15 or so, and use a spline curve algorithm (or plain old french curve by hand) to interpolate the elliptical path that the pendulum follows. What I mean is, I am not sure if the actual distorted path is a perfect ellipse, it may be even more messed up than that but it is a decent first order approximation because we can at least see that the circle can only satisfy at most two points.

    Attached Files:

    Last edited: Jun 24, 2009
  6. Jun 24, 2009 #5


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    What attached JPEG? Did we forget something? :wink:

    EDIT: Ah, there it is.
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