4. Mathematical Formulae in Action

Mathematical formulae are a very important and very common class of algebraic equations. You have already seen many mathematical formulae, e.g. area of a circle $=\pi r^2$, or the formula for the amount $A$ resulting from principal $P$ at interest rate $r$ % after $n$ years ( $A=P(1+\frac{r}{100})^n$). Writing down such formulae (equations), is an important goal in modern science and engineering. For example, it is possible to write down a formula which can be used to calculate how much a bridge will sag when a truck of a given weight stands on it. This formula will of course be very complicated- but bridge designers need to have such formulae in order to decide whether the bridge is safe or not. Similar examples can be found in every branch of science and engineering. We will not study all that here (for that you will have to go to science/engineering colleges)¹; here we just want to make a small beginning.

In this chapter we are going to see some very simple objects and phenomenon and learn to write appropriate formulae for them. We will also learn how to use the formulae, and some new phrases that are commonly used in science and engineering in connection with writing such formulae.

We will begin with an example.

0.1 Rubber Band Weighing Machine

Figure 4.1: Rubber band weighing machine

Let us perform a computer experiment. Figure 4.1 shows a rubber band suspended from a nail. Its length is 10 cm. Depress your mouse button on the 10 gm weight, then move it close to the rubber band and release it.. This will cause the weight to get attached to the rubber band. But as you do so, like a real rubber band, our rubber band will also extend! You should see that its length has increased to 11 cm. Now click on the 10 gm weight and you will see that it returns back to its place. Now attach the 20 gm weight to the rubber band by clicking on it and moving the mouse. You can see that this time the rubber band extends by 2 cm so that its total length becomes 12 cm.

From these facts you might speculate that attaching a weight of 30 gm might increase the length to 13 cm - and in fact you would be right, (unless of course the rubber band breaks)! (Try it!) This observation can be noted in the form of an equation that characterizes the behaviour of the rubber band:

$$l=w/10$$ (1)

Here $l$ denotes the change in length of the rubber band measured in cm, and $w$ the attached weight in gm. This formula will work for any value $w$ (so long as it is reasonably small, otherwise the rubber band will break)- given any $w$ you can calculate the increase in length $l$ caused by it. In fact, you can use the formula in the reverse direction too. Suppose you attach an unknown weight and the increase in length is 1.6 cm. Substituting this into the formula we get $1.6=\frac{1}{10}w$, from which we can conclude that the attached weight $w$ must be 16 gm. You can make your own weighing machine in this manner.² As you can see, the formula is very useful.

Can you guess the weight of the rightmost weight which is marked $x$? All you need to do is to attach it to the rubber band. Try it! You can change the unknown weight by pressing on the ``new unknown weight'' key. Can you find the unknown weight now?

0.2 Chapter Outline

The goal of this chapter is to study the general process of developing formulae of the kind described above. What we will study in this chapter will be useful not only in understanding the rubber band formula better, but also other formulae which will be of interest in various branches of science and engineering and even economics. There are essentially two parts to this:

1. Knowledge of some general laws regarding the process or object being studied. In the case of the rubber band problem, this means that we need to know some law from physics which says how rubber bands behave when weights are attached to them.
2. Experimental data. In the case of the rubber band problem, the experimental data that was given to us consisted of the observations that the rubber band stretched by 1 cm when a weight of 10 gm was attached and by 2 cm for 20 gm.

In what follows, we will study the two steps given above in the context of the rubber band problem and also other problems from physics and chemistry. You might wonder why we are studying physics and chemistry in a mathematics text book. This chapter is really not about physics or chemistry; it is about the process of deriving and manipulating mathematical formulae. We are simply using formulae from different sciences as examples.

1 Rubber Band Example (contd.)

The formula we wrote down for the rubber band is based on a general principle named after its discoverer, Robert Hooke. Here is a statement of the version of his principle meant for rubber bands with weights attached.

Theorem 1 (Hooke's Law for Rubber Bands)   Let $l$ denote the increase in the length of a rubber band when it is suspended from one end and a weight $w$ is attached to the other end. Then $l=kw$ where $k$ is a fixed number. The number $k$ may be different for different bands; for any given band it is fixed, independent of the weight attached to it.

So we know from Hooke's law that the equation describing the increase in length of our rubber band must be $l=kw$ where $k$ is some number that we need to determine. How do we determine $k$? For this we need some experimental data.

1.1 Use of experimental data:

In the problem as stated in Figure 4.1 we were given two observations. Let us just consider the first observation: when the applied weight is 10 gm, the extension is 1 cm. Now let us write down what we know about the rubber band:

Hooke's Law:

$l=kw$.

Observation:

$l=1$ when $w=10$.

From this it follows that $1=k\cdot 10$, i.e. $k=\frac{1}{10}$. Substituting this value of $k$ into what we know from Hooke's Law we get

$$l=\frac{1}{10}w$$

which is identical to Equation 4.1.

You may wonder whether we need the second observation given in the problem (extension is 2 cm when 20 gm are applied). This observation is in fact not necessary. Given that the stretching of the rubber band is governed by Hooke's law, a single experimental observation is sufficient to determine $k$ and equation. In fact, once $k$ is known, we can use our equation to predict the extension when 20 gm are attached. For this we need to substitute w=20 in the above equation:

$$l=\frac{1}{10}20=2$$

The extension is indeed 2 cm as per our second observation, and hence our prediction is correct.³

1.2 Exercise

1. Suppose I have a rubber band whose length increases by 1.5 cm when I attach a 20 gm weight. What will the increase in the length be if I attach a 60 gm weight?
2. For the same rubber band, what weight do I need to attach in order to increase the length by 6 cm?

You should first do the calculation and then check your answer by attaching weights to the rubber band in Figure .

Figure: Rubber band for exercise

2 Language used in expressing Laws

We will study a phrase that is commonly used in describing scientific laws.

Definition 1   Let $x,y$ be variables. We will say that $x$ varies in proportion to $y$, or varies as $y$, or is in proportion to $y$ if $y=kx$ where $k$ denotes some fixed number.

Notice that the definition does not say what $k$ needs to be. Any value is acceptable. The only requirement is that there needs to be a single fixed $k$ for any two variables $x$ and $y$ that we are claiming to be in proportion.

Using this phrase we will write Hooke's law as:

Theorem 2 (Hooke's Law for Rubber Bands)   The increase in the length of a rubber band suspended from one end varies in proportion to the weight $w$ attached to the other end.

There are several other examples where variables vary in proportion. For example, let $P$ denote the price of $q$ kg of sugar. Then we know that $P=cq$ where $c$ is the cost per kilogram. Therefore we can say that $P$ varies in proportion to $q$. As another example, let $d$ denote the distance travelled by a car moving at a fixed speed $S$ in time $t$. Then clearly, $d=St$. Since we have said $S$ is fixed, the conditions of definition 1 are satisfied. Thus we can say that the distance covered varies in proportion to the travel time so long as speed remains constant. As a final example, circumference $C$ and the radius $r$ of a circle are related as $C=2\pi r$. Thus we have $k=2\pi=$ constant as in definition 1 and thus we can say that the circumference of a circle varies in proportion to the radius.

Now let us give some examples of variables that do not vary in proportion. The area of a circle does not vary in proportion to its radius (in fact it varies in proportion to the square of its radius). For fixed interest rate and duration, the interest earned varies in proportion to the amount invested if the interest is simple, but not if the interest is compound.

We note in conclusion that the phrase ``in proportion'' is useful because it is applicable to many commonly occuring pairs of variables. By using this phrase we can compactly describe the relationship between the variables. This is most easily seen in the statements of Hooke's law - the second statement is much more compact. Use of this phrase in Hooke's law is important for another reason: it reminds us that the relation between the weight and the extension is similar to the relation between many other pairs of commonly occuring variables (e.g. price and quantity, perimeter and radius).

Note on ``variable'' and ``unknown'':

Terms appearing in the formula (such as $w$) are typically called variables. In the previous chapters we used the word unknown to denote terms appearing in equations. We use different names only to signal our intention: the term $x$ appearing in equation 1.2 was representing a number that is fixed but which we do not know. Whereas when we wrote Equation 4.1 we intended that we will be correct to substitute different values in place for $w$. Hence we use the term variable rather than unknown. However, for mathematical purposes there is no distinction - we can manipulate all equations in a similar manner. $d=St$ provides a better example - $d,t$ are variables, while $S$ is an unknown constant. Similarly $l=kw$.