What's going on when the pendulum swings?

[Hernik]

Hi. Hope someone can help me with this.

This week I was asked to write a children's page on natural science. One of the things I need to do is describe a simple experiment, that can be done at home. So I decided I would describe how you can time the swing of a weight in a line, and how the time for a full swing back and forth will be the same regardless of the amplitude of the swing. The good old Galilei thing. (I realize it is only an approximation and only works at relatively small angles) This part was easy. But then I have to describe "Why". Why does it work like that. And when I think of it I realize I do not know.

I tried to search for an answer on the internet but without luck. My intuitive guess on what is going on is that inertia is being converted into potential gravitational energy and then back again. And what is lost in this dance of two forces is due to friction. But why is the time for a swing the same regardless of the distance travelled. Can this be explained simple and in words?

- Henrik

 

[Cleonis]

But why is the time for a swing the same regardless of the distance travelled.

For background I think it is helpful to generalize.
The insight can subsequently be focused on the pendulum case.

It is a general property of harmonic oscillation that the period is independent from the amplitude.

You get a harmonic oscillation when the restoring force is proportional to the displacement from the equilbrium point.

Let's say you have two identical setups for a harmonic oscillation, A and B. You set them both in motion, Giving B twice the amplitude.

(In the case of a pendulum the oscillator is the pendulum bob.)
- At the extremal points of the oscillation oscillator B is subject to twice the force.
- At the extremal points of the oscillation oscillator B undergoes twice the acceleration.
- At every point of the oscillation oscillator B has twice as much velocity as oscillator A.
- Oscillator B has to cover twice the distance (because the amplitude is twice as large) but it takes the same amount of time as for oscillator A because oscillator B has at every point twice the velocity.

A graphical way of illustrating this is to first point out that if you plot position over time as a line, then the steeper the line the larger the velocity.

If it is granted that A and B are both in harmonic oscillation then both are represented by a sine function, but the plot for B is everywhere twice as steep as the plot for A.


Presumably explaining why a proportional restoring force give rise to a harmonic oscillation is beyond the scope of what you want to communicate. But perhaps if your intended audience is willing to grant you that for both A and B the plot is a sine function (suitably proportioned) then maybe what you want is feasible.

 

[Hernik]

Thank you so much. This was very helpful!

- henrik