About Formulas

At school you get to know a great many formulas. Your teacher explains a relationship to you, and at the end there is a formula for you. Your teacher is then happy. Many YouTube videos on technical topics work in exactly the same way: a very weak explanation of a relationship is followed by the formula, as if this were the goal of an explanation.

A formula on its own is worthless. It only makes sense in a technical context and when you really need it. Formulas are important for describing the world around us mathematically. Today, however, all the relationships you calculate in your studies have already been described with formulas. Mathematics is much further ahead than you need for your formulas. You can look up any formula at any time. So why are formulas (still) the goal at the end of an explanation?

Formulas define a relationship with the help of mathematics in the form of an equation. We can say a lot about the relationship from an equation. First of all, it contains an equals sign, so both sides of the equation are equal. If we need a larger value on the left side of the equals sign, we must also insert a larger value on the right side. For example, 2 = 2 holds; this is a true statement. If we increase the left side to 3, there is only one true statement for it: 3 = 3. The right side must also be increased by the same value.

If a formula has the form

\[ y = \frac{a}{b} \]

then y can be increased if a becomes larger or if b becomes smaller. Whether a quantity is in the numerator or the denominator is always important. If y is to become smaller, either b must become larger or a must become smaller. So the formula also shows how a quantity acts. Quantities in the numerator increase the numerical value. Quantities in the denominator reduce it.

For a pumped-storage hydroelectric power plant, water is pumped up into a mountain lake at night. During the day, the water flows back through the turbines into the lake in the valley. During the day, an energy of W = 1 MWh is needed. You are to design this plant. How do you proceed?

You quickly realise that you need formulas for the design. So you look for formulas for energy. You find many formulas on the internet:

1. Thermal energy W_THE=…

2. Kinetic energy W_KIN=…

3. Potential energy W_POT=…

4. Electrical energy W_EL=…

5. Chemical energy W_CH=…

and much more …

You will find articles and formulas for all forms of energy. Apparently there is no single formula with which energy can be calculated. Each formula is only valid for a certain type of energy, and then only for a certain relationship or context. Formulas are often described for a particular geometry in the application, and even a slightly different setup gets a different formula. So the art does not lie in calculating a result with a formula – apps will soon be able to do that on their own without you. The challenge lies in finding the right formula for your application.

Let us look at potential energy. It describes an energy that is determined by the gravitational force, i.e. the Earth's gravity. This form of energy is important for pumping water up or for calculating how much energy is released when it flows down. Let us take a closer look at it as an example. Let us put on the glasses of an engineer who does not just insert numbers and calculate a result, but who understands the formula.

For mechanical forms of energy in which a force acts along a path, the following generally applies:

\[ W = \int F\,ds \]

Intuitively, we have to invest energy to lift an object. Energy is released, e.g. for an independent movement, when the object falls down. So here a force acts along a path, and the path is the change in height h. The following applies:

\[ W_{\mathrm{Pot}} = \int F_g\,dh \]

We can describe the gravitational force with the formula F_g = m ∙ g. It is valid in the context that masses are moved in a gravitational field along the field lines of the field. On the Earth's surface, the field strength g has the constant value

\[ g = 9.81\,\frac{\mathrm{m}}{\mathrm{s}^2} \approx 10\,\frac{\mathrm{m}}{\mathrm{s}^2} \]

On the Moon the value of g is smaller, on the Sun it is larger; it depends on the mass of the planet. If we move away from the Earth, the value of g also decreases; it depends on the distance to the centre of the planet. When in doubt, you have to read up a lot to apply this supposedly simple formula correctly. If the gravitational force is inserted, the following applies:

\[ W_{\mathrm{Pot}} = \int (m \cdot g)\,dh \]

The parameters m and g are independent of the height h. Therefore the integral can be simplified to W_POT = m ∙ g ∙ h. Solving a real problem is different from solving an exam problem. Formulas are used for both. Use your studies to question formulas with which a problem can supposedly be calculated.

How do you handle formulas correctly in your studies?

1. You are often allowed to use a formula sheet in the exam. Note the context in which each formula is valid, so that you use the right formula in the exam.

2. If you understand what happens when the parameters of the formula change, you can judge whether you have the right formula for the problem. If, for the design of the pumped-storage plant, you find energy formulas with temperature, charge or radiation intensity as parameters, they obviously do not describe your problem. Research possible valid formulas, get a feeling for the parameters of the formula and their effect in the formula, and thus make sure that you have found the right formula.

3. Check which geometry the formula applies to. There may be a special formula for your geometry.

4. Never insert numbers into a formula without thinking and then declare the result valid.