## THE CLOCK

**Editor's Note**: This math project by**George Benz ('16)**provides a mathematical and scientific explanation to the functions and shapes of a clock.**The Art of Horology Part 1: Work **

How do watches tick with such incredible precision? How were tinkerers of the 1800’s able to produce miniature machines that could tick with such persistence that they would only lose seconds each day? They did it with the magic of gears and the immense amount of work the gears can do with a minimal amount of energy.

Perhaps the single most important part of a watch is the balance wheel and the lever escapement. It is the heart of the watch. This is essential to the watch as it regulates the amount of work outputted from the main spring. It makes sure that only a certain fraction of energy stored in the mainspring is released into the gears, an equal amount each time to make sure the ticking stays constant.

In order to learn how a watch works and to better demonstrate it, I recreated the most essential parts in a program known as Algodoo, which allows me to create 2 dimensional shapes, gears, spring, hinges, etc. in a functioning physics simulation.

This is how it works:

Perhaps the single most important part of a watch is the balance wheel and the lever escapement. It is the heart of the watch. This is essential to the watch as it regulates the amount of work outputted from the main spring. It makes sure that only a certain fraction of energy stored in the mainspring is released into the gears, an equal amount each time to make sure the ticking stays constant.

In order to learn how a watch works and to better demonstrate it, I recreated the most essential parts in a program known as Algodoo, which allows me to create 2 dimensional shapes, gears, spring, hinges, etc. in a functioning physics simulation.

This is how it works:

While that explanation was a bit wordy and complicated, the main point of this mechanism is to have the energy in the mainspring (the hairspring that is wound up by the twisting of the stem) let out in equal fragments. All the watch is really doing is recording the energy released, and that means the energy flow must be constant. Also it should be noted the above process is done, and repeated, around 5 times per second. The constant taping of the lever arm against the teeth of the mainspring causes that tick tock sound. But the question still stands, how can something keep powered for days off of a small hairspring?

The law of conservation of energy states that the change in total internal energy of a system equals the added heat, minus the work performed by the system1. To us this means the energy stored in the mainspring is converted into work. While work is much more complicated than this, simply put, work is basically force times displacement. How much force you are putting in to move something a certain distance.

Now what is force? Force is the mass of an object times the acceleration. If you kick something that weighs a little it will go very far since its small mass, but if you try kicking a building it won’t move at all. This applies to the very small gears in watches as well. There incredibly tiny mass allows very little force to be applied to them. Also, the acceleration of the balance wheel and lever escapement are very little. Very small accelerations and masses make it so the force needed to make one tick is incredibly minute.

Now back to work. The work stored in the hairspring is quite small, so the watch makes up for it by having minimal force needed. To add on to this, the displacement of the gears turning is TINY, each one turns around 1 millimeter. Both of these factors combined make it so one tick of the watch requires minuscule work, which allows it to run for so long.

The law of conservation of energy states that the change in total internal energy of a system equals the added heat, minus the work performed by the system1. To us this means the energy stored in the mainspring is converted into work. While work is much more complicated than this, simply put, work is basically force times displacement. How much force you are putting in to move something a certain distance.

Now what is force? Force is the mass of an object times the acceleration. If you kick something that weighs a little it will go very far since its small mass, but if you try kicking a building it won’t move at all. This applies to the very small gears in watches as well. There incredibly tiny mass allows very little force to be applied to them. Also, the acceleration of the balance wheel and lever escapement are very little. Very small accelerations and masses make it so the force needed to make one tick is incredibly minute.

Now back to work. The work stored in the hairspring is quite small, so the watch makes up for it by having minimal force needed. To add on to this, the displacement of the gears turning is TINY, each one turns around 1 millimeter. Both of these factors combined make it so one tick of the watch requires minuscule work, which allows it to run for so long.

**The Art of Horology Part 2: Gears **

In order to convert this quick movement of the lever escapement and balance wheel into seconds, a ratio must be made between the gear attached to this movement and the gear that will move the second hand. The second hand gear must move at 1 revolution per minute and the lever escapement gear moves at around 10 revolutions per minute and has around 10 teeth (this value is different between watches). How many teeth should we give the second hand gear?

To find this out I have given an example of two gears with different teeth amounts. The gear on the left has 25 teeth while the gear on the right has 50. As you can see when you turn the gear on

To find this out I have given an example of two gears with different teeth amounts. The gear on the left has 25 teeth while the gear on the right has 50. As you can see when you turn the gear on

the left half a revolution the gear on the right turns a quarter of a revolution. That means that a gear teeth ratio of 1:2 will results in a revolution difference of 2:1. They are inversely related.

To even further demonstrate this I have added a 100 teeth gear to the 25 teeth gear. As I turn 25 teeth gear half a revolution, the large 100 teeth gear only turns an eight of a revolution. So 1:4 teeth ratio will equal a revolution difference of 4:1.

To even further demonstrate this I have added a 100 teeth gear to the 25 teeth gear. As I turn 25 teeth gear half a revolution, the large 100 teeth gear only turns an eight of a revolution. So 1:4 teeth ratio will equal a revolution difference of 4:1.

So for the original question of how many teeth the gear of the second hand should have, we just need to set up a simple ratio. The inverse of revolution difference (1/10) should be equal to the teeth difference X/10 (X is the number of teeth in the second hand gear). So 10/1 should be equal to X/10. If we solve this we will see that in order to convert 10 revolutions per minute to 1 revolution per minute we will need the second hand gear to have 100 teeth. These fundamentals of watches is what allow them to record the flow of energy into simple measurements like seconds, hours, and even days.

Observationally this can be answered, but what about experimentally? What exactly is a revolution? One could say it is when a point of the circumference travels 360 degrees back to its original position. One could also say that the teeth around the gear are actually just the circumference of the gear in the unit of teeth. Turning the 25 teeth gear half a revolution is like taking 180 degrees of the 25 teeth circumference, which is 12.5 teeth, and then turning the 50 teeth gear that same amount. But now on the 50 teeth gear being turned 12.5 teeth only makes it so the gear revolves 90 degrees, since 12.5 teeth out of the total 50 teeth circumference is only a quarter.

Gears are really just differences in circumference, and how imposing an arc of one on another, will create differences in angles.

Observationally this can be answered, but what about experimentally? What exactly is a revolution? One could say it is when a point of the circumference travels 360 degrees back to its original position. One could also say that the teeth around the gear are actually just the circumference of the gear in the unit of teeth. Turning the 25 teeth gear half a revolution is like taking 180 degrees of the 25 teeth circumference, which is 12.5 teeth, and then turning the 50 teeth gear that same amount. But now on the 50 teeth gear being turned 12.5 teeth only makes it so the gear revolves 90 degrees, since 12.5 teeth out of the total 50 teeth circumference is only a quarter.

Gears are really just differences in circumference, and how imposing an arc of one on another, will create differences in angles.

1. http://en.wikipedia.org/wiki/Work_%28physics%29#Work_and_energy