Different people will answer this question in different ways. Some people believe that all scientific research benefits everyone. At the other extreme, other people believe that most scientific research is harmful. I believe that most scientific research benefits somebody but, at the same time, some of it may have the potential to cause harm. Let’s… Continue reading 18.18 Who benefits from scientific research?

In post 18.15, we saw that the number e raised to the power x (see post 18.2) is defined by the series (post 17.11)           ex = 1 + x +x2/2! + x3/3! + x4/4! + x5/5! + … Here, for example, 5! = 5 × 4 × 3 × 2 × 1. In post… Continue reading 18.17 Euler’s relation, oscillations and waves

It may seem that this post is about very abstract mathematics. But the ideas explained here are very useful for explaining oscillations and waves (post 18.11), especially when combined with the concept of the number e (post 18.15). Counting is like moving along the line in the picture above – from left to right. When… Continue reading 18.16 The square root of minus 1 and complex numbers

I wrote about exponential growth and decay in posts 16.5 and 16.6; I want to return to this topic again to introduce some ideas that I will use in later posts. Let’s invent a number called e (it was really invented by Jacob Bernoulli in 1683); we will define it raised to the power x… Continue reading 18.15 More about exponential growth: the number e

Before you read this, I suggest you read posts 18.10, 18.11 and 18.12. The picture above shows waves on the sea. Their shape is much more complicated than the simple sine and cosines waves (see picture below) that I have used to represent waves in previous posts (like post 18.10). Can we use the ideas… Continue reading 18.14 Wave shapes – Fourier series

When an object moves in air, it must push the air away from it; it changes the motion of the air and, by definition, so exerts a force on it (post 16.13). Since pressure is simply force divided by the area on which it acts (post 17.5), the object increases the pressure in the direction… Continue reading 18.13 Sound

Before you read this, I suggest you read posts 18.8 and 18.10; it might also help to read post 18.11. Let’s suppose a string or wire is fixed at A, is stretched tight by a force T (a tensile force, post 16.49) and is then clamped at B. This wire is shown in black. Now… Continue reading 18.12 Vibrating strings: violins, guitars and pianos

In post 17.12 we saw how motion in a circle can be described, although some ideas on this subject appeared earlier in post 16.14. We met the simple harmonic oscillator in post 18.6 but some of the equations used were derived more simply in post 18.7. Finally, in post 18.10, we looked at wave motion.… Continue reading 18.11 Motion in a circle, the simple harmonic oscillator and waves

If you are not familiar with measuring angles in radians, I suggest you read post 17.11 before this. The picture shows what happens when we drop a stone into a pond: the stone disturbs the water and waves spread out from the point of disturbance – the source of the waves. Eventually the waves will… Continue reading 18.10 Waves

Before you read this post, I suggest you read post 17.17. In post 18.8 we saw the problems that unwanted motion of bridges could cause. Another example of unwanted motion is the sudden jolt that we experience when a vehicle goes over a bump. Fortunately, most vehicles are fitted with shock-absorbers to remove the effect… Continue reading 18.9 Damping and muscles