Pendular Motion


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Physics - Rotational Method

The spirals, ellipses, and figure eights are graphical records of the swinging pendulums. Lift the handle to start this row of pendulums swinging, and then watch as they move in and out of a repeating series of mesmerizing patterns.

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A Pendulum Clock At first glance, the giant see-through-erector-set-like structure seems to be a realization of one of Leonardo da Vinci's mechanical inventions. Chaotic Pendulum When you set these pendulums swinging, the motion of each one affects the others. Drawing Board Draw hypnotically flowing patterns with a swinging table, and watch friction cause the patterns to slowly shrink along a spiral path. Break the motion into small time steps. During each time step:.

Pendulum Lab

But this doesn't work with the pendulum. The tension in the string of a pendulum is clearly a force of constraint. Sure, the direction of this tension force is in the same direction as the string but the magnitude changes to whatever value it needs to be to keep the mass at the same distance from the pivot point.

This means that in order to make a numerical model for a pendulum, you need to use a trick. There are three different ways you can model the motion of a pendulum. I have looked at these methods before, so let me just give a short review. Notice that the title of that post is "a third way.


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  • If you assume the mass is confined to move in a circular path, then you can reduce this to a one dimensional problem with the angle of the pendulum as the only variable. The only force that changes this angular position is the angular component of the gravitational force.

    There is a simple solution to this differential equation by assuming a small amplitude of oscillation and thus a small angle.

    The problem with the pendulum motion is that the tension is a constraint force. Well, what if we make it a deterministic force?

    Modeling a Pendulum's Swing Is Way Harder Than You Think

    If the string is replaced with a very stiff spring, it should be an easier problem. This method can work fairly well. Here is a numerical model that displays the angular position for both method 1 and 2. Just click the "play" button to run this. If you want to change some of the code and you probably should , I have left comments to indicate which things you could change. Don't worry, you won't break anything. Just click the "pencil" icon to switch to the code mode to edit. Really, you should play around with the values for mass, spring constant k and time step dt to see how well this model agrees with the differential equation.

    The pendulum

    Hint, try looking at both models to see which one is better at conserving energy. Yes, you can consider that a homework assignment if you like. I can use the usual numerical model method if I can find an expression for the tension during each time step. Let's take a look at the forces on the mass during a swing. I already know the direction of this tension force—it has to be in the same direction as the string because strings only pull.

    But what about the magnitude?

    Pendulum Waves

    In that case, I can add up the forces in the direction of the string I will call this the r direction. With the net force in the r-direction, I know this must also be equal to the mass of the object multiplied by the acceleration in the r-direction. Since the object is moving in a circle with a radius of L and a speed of v , it will have a centripetal acceleration towards the center of the circle in the direction of the tension.

    Factors and parameters

    Now I have an expression for both the magnitude and direction of the tension force based on the angle and the velocity. With this, I can just add a line into my numerical calculation loop and determine the vector value for the tension force. After adding this to the gravitational force, I can use the momentum principle—that should work. Here is this method as a numerical calculation.

    I have again included the solution for the differential equation for comparison. Again, click the play button to start this. Also, you should play around with the code. Why does anyone need to use this third method for the motion of a pendulum? Really, it's all about introductory physics courses.

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