Where would you start to design a watch movement? It is a tricky question, and to be honest, I am not too sure right now. But what I thought would be fun would be to see if I could make a reasonable stab at the problem.

I spent five or six blogs diving into steel and the nuances of that marvelous material. Then looking at a few other materials used in watchmaking, including the watch crystal and Swiss Super-LumiNova. These are manageable subjects to understand as I can qualitatively write about them. This is all very nice, but we are more interested in how the mechanics of a watch work. After all, it is the mechanics in the back of a watch that keeps time. There is no need for the stainless steel case or the watch crystal unless we have a watch movement inside the case.

#### What Question To Answer Next?

The critical question is, how does a spinning balance wheel keep time, and what are the forces at work? How are the sizes of each wheel chosen? And then how are the shape and dimensions of the teeth on each gear determined? These questions may seem very esoteric, but the design of each of these components impacts how your watch keeps time.

I aim to explore each of these questions, and a few more, over a series of blog posts. Each blog will focus on a specific component or module of a watch movement. The intent is that over time, the collection of blog posts will slowly unravel the mechanics of a watch movement. Please do not hold your breath because I expect this journey to take almost a year or so.

You may be thinking that there are plenty of resources out there that explain the intricate workings of a watch movement. This is true. But my aim here is not to merely explain in words what is going on. I am going to approach explaining a watch movement from a different angle, from first principles. In last week’s blog, I expounded on the process of deconstructing problems and questions to first principles, so that is what I am going to do for a watch movement. For every component or working assembly, I will look at the fundamental principles that govern how it works and then, with some mathematics, illustrate the critical design criterion behind the component.

#### Where To Start?

If you take the back off a watch, the obvious observation is that the vast majority of the active components rotate in some way, shape, or form. That may seem a statement of the obvious, but if we dissect the problem from first principles, then we need to get into the obvious.

#### There Will Be Mathematics

If everything, from the barrel to the balance wheel and all the gears in between rotating, we better understand how to describe a rotating body accurately. To describe the movement of any physical body, we are going to have to use mathematics. I am sure that will have sent many of you to hover over the “close” icon on the webpage. I would ask you to read on a little further.

#### Do Not Be Scared!

My journey with mathematics was not an easy one. I was not always a fan of the subject. The breakthrough for me was when I realized that mathematics is just another language. It could be described as the language of science. In engineering, it is used to describe the world around us in a quantifiable manner that allows us to test possible outcomes in the virtual world before creating them in the real world.

I appreciate that mathematics is not everyone’s cup of tea. To hopefully make this more approachable, I will include all the detailed workings so that there is a complete and accurate for those interested. More importantly, though, I shall use the process to illustrate the detailed design considerations for each component. I shall translate the mathematics into understandable concepts. More accurately, I should probably say I shall attempt to make these processes more understandable. You will need to be the judge of my success in this regard. If you can understand the physical principles that impact the component’s design under discussion at the end of the blog, that would be a good result. This goal is a lofty objective, and I hope I am up to it. It will be for you to decide.

#### Everything Rotates – A Rotating Body

If everything of importance rotates in a watch movement, then we need to describe rotating bodies accurately. Today, I will look at circles, their properties, and how we use them to describe rotary movement in mathematics. This understanding will allow us to create a mathematical model of the various components, which will, in future blogs, inform our understanding of the design of multiple components.

#### Let’s Start With Linear Motion

Linear motion is a fancy name for going in a straight line. Driving a car in a straight line is linear motion, which gives rise to the measurement of speed. In the world of engineering, this is generally referred to as velocity. We can measure velocity in various units, depending on the quantum of what we are measuring. We are most familiar with miles per hour or kilometers per hour, depending on our country. To calculate the velocity, we measure the distance traveled and divide that by the time taken to travel the measured distance. This calculation will provide us with the average velocity over the distance traveled.

We can represent this in an equation. If the distance is defined as “S” the time taken to travel the distance is “t” then the velocity “V” can be calculated as follows:

**V = S/t**

This equation is for something traveling in a straight line. But how can we measure the speed of something rotating?

#### Going Round In Circles

We intuitively understand that going round in a circle is different from going in a straight line. The question is, where do we start with understanding the differences? The most obvious place is looking at a circle and understanding how it is defined and its defining features.

Every circle has two defining measurements. The first is the diameter of the circle, and the second is the circumference of the circle. For any circle, these two measurements are linked by a physical constant called Pi. So, for any given circle, where the diameter is “D,” and the circumference is “C,” then Pi can be calculated as follows:

**C/D =Pi **

Or, to explain this another way, if you divide the circumference of any circle by the diameter of the same circle, the answer will be Pi.

#### The Physical Constant

What do I mean by a physical constant? A physical constant is one that never changes. In this case, it means the number, Pi, never varies. Pi is particularly interesting as it cannot be expressed as a finite decimal number. In most cases, an approximation is used, 3.14 or 22/7 are the most common approximations, but Pi has no limit to the number of decimal places to which it can be calculated. Pi cannot be represented as a fraction nor expressed as a finite decimal and is referred to as an irrational number.

For our purposes, it is sufficient to know that if the circumference of a circle is divided by its diameter and the answer does not equal Pi, it is not a circle. In mathematical equations, Pi is represented by the greek letter “𝝅.”

In mathematics and engineering, it is more common to use the radius of the circle in calculations. The radius is the distance from the circle’s center or the “focus” of the circle to the circumference. The radius is, therefore, half the diameter of the circle; this results in the most commonly seen equation for the fundamental properties of a circle can be expressed as

**D=2𝝅r **

where “r” is the radius of the circle.

#### Segments of A Circle

By understanding the relationship between the radius of a circle and the circumference of a cricle, we can calculate the length of an arc on the circumference of a circle. For example, if we are looking to calculate the arc on a watch face between positions 12 and 3, we would most commonly describe this as 90 degrees out of the 360-degree circle. If the watch face has a radius r, the calculation is, therefore:

**Length of arc = 2𝝅r * 90/360**

But what if we know the length of the arc rather than the angle? Instead of defining the angle of an arc in degrees, what happens if we define the angle by the ratio of the length of the arc and its radius?

Let’s consider the diagram below where the arc has a length “l” and a radius “r,” the angle”⍺”swept by the radius between the beginning and end of the arc can be defined as:

**⍺=l/r**

The benefit of this may not be immediately obvious, but if “l” is equal to the circumference “C” of a circle, then we end up with the equation:

**⍺ = C/r**

We also know from above that C=2𝝅r. And by substituting this into the equation above for C, we can show that **⍺ = 2𝝅** when l = C.

#### A New Way To Measure Angles

The benefit of this is that is if the angles are defined in this manner, then the linear distance can be calculated by merely multiplying the angle by the radius. This measurement of an angle is referred to as “Radians” and, as you can see, 2𝝅 radians equals 360 degrees.

For example, the hand on a watch face will sweep out 𝝅 radians as it moves from 12 to 6. If the hand is “r” long, then the distance traveled at the tip of the hand can be calculated as:

Length of arc = angle swept (in radians) x length of the hand

In this case, the answer would be 𝝅r. For a 20mm hand, the tip will move a distance of 62.8 mm.

#### Speed Of Rotation

Now we have a method to calculate the distance traveled along the arc of a circle. If we can time the interval for the movement to happen, we can calculate the speed with which the arc is swept.

Let’s use the example of a 20mm second-hand sweeping around the watch face. It will travel 2𝝅 radians in one minute. The distance traveled is, therefore:

2𝝅r = 2 x 3.14159 x 20 = 125.7 mm per minute or approximately 2 mm per second.

It is nice to know how fast something moves around an arc, but unfortunately, it is not very useful. It is most beneficial to know the angle swept out by a radius over a given period when dealing with rotation and is referred to as the angular velocity.

For our example above, the second hand sweeping out 2𝝅 radians every minute the angular velocity is 2𝝅 radians per minute. This is how we need to think about rotating gears and wheels’ motion in terms of their angular velocities.

#### These Concepts Are The Foundations

These are the fundamentals that we will build our models of all the rotating components in the watch movement. Whether we are talking about the movement of the balance wheel, the movement of the wheels and gears, and even the shape of the gear teeth, these understanding angular velocities and the fundamental aspects of circles will be critical to understanding the design.

I hope this has been enjoyable to read, but most of all, I hope that it has helped you to understand a little about circles and how to describe a rotating disc. I am keen to improve the explanation so that hopefully, as we start to dissect the watch movement in more detail, as many people as possible can understand what is going on. If there are confusing areas that do not make sense, please let me know in the comments below. I will do my best to expand these sections and make them more easily understood.