As with many of the old favourite playground staples, swings offer us the chance to experience many aspects of Physics. Every child loves that weightless feeling that you get at the height of the swing, although the small gulp that goes with overcooking it and letting the chains go slack is just your body’s way of letting you know that we probably weren’t meant to fly!
Swings are great for looking at energy changes, because we can see and feel the effects of everything we do with our bodies providing the mass. The only problem is that our legs get in on the action to provide extra energy to keep us going. If you can keep still you’ll feel the potential energy of raising the swing up change to kinetic energy once you release it, and then this gets turned back into potential energy at the other end of the arc. The process then repeats, but small energy losses like noise and heat to the connection between the swing and the frame, as well as air resistance, mean that you’ll never go as high as the first time. Eventually you come to a stop when all the energy has dissipated.
Pendulums work in exactly the same way, but because the system is fixed and there’s no person wriggling about and trying to go higher, you can make more accurate predictions on the behaviour. The famous thinker Galileo was the first to notice (when chandeliers swung!) that whether a pendulum has a big or a small swing the length of the swing takes the same amount of time (although this is only true for small angles, not giant swings like the sort that make you gulp!) and it set him thinking about time, and accurately setting a clock.
So if the angle you release your pendulum from doesn’t make a difference to the period, which is what the time between swings is called, then what does? The first thing you can try is changing the mass, but here too (unless you get something that really ramps up the air resistance part of the energy loss) there is no difference, as the energy conservation works just the same. Only one thing left to try then, and that’s the length of the chain you attach the pendulum to.
Finally! Something that makes a difference! The length of the chain is directly related to the period of the swing, which means that you know if you have a shorter chain the swings will be faster than if you have a longer chain. All that remained was for someone to find the exact relationship. Now, this can’t have been an easy task so let’s look at what they started with. They knew that potential energy was transferred, so that implies that mass and gravity are important, but we know that mass makes no difference, so we can discount that. And length is directly related to the period. In fact, the relationship is fairly complex:
So if we want to know how long our pendulum should be so we can time a second, we can just use the equation
That’s quite long, for a clock, isn’t it! Imagine trying to carry that about! Of course they never did, but the Grandfather clock was the direct result, and you can see why it has to be so tall. Oh, and remember that energy loss, well that’s why you need to wind it up every now and then, to keep it going.