You step on your bathroom scales to find your body mass. The idea is that gravity acts on your mass to exert a force that depresses a platform. The depression of the platform depends on the force and, hence, on your body mass. So, to find your body mass, you need to measure the depression… Continue reading 24.8 Passive damping – bathroom scales
Before you read this, I suggest you read posts 24.5 and 24.6. In post 21.1 we saw that an electrical circuit that consists only of a capacitor discharging through an inductor is a simple harmonic oscillator in which the charge, Q, depends on time, t, according to the equation Q = Q0eiωt (1) (equation B4… Continue reading 24.7 Driven oscillator
Before your read this, I suggest you read post 24.5. In post 24.5, we saw that the displacement, x, of a damped simple harmonic oscillator is given by equation 1. Here t represents time, dx/dt represents speed and d2x/dt2 represents acceleration. As in post 24.5 we are considering an object that oscillates along a line,… Continue reading 24.6 Damped simple harmonic oscillator
In post 18.24 we saw that we could represent the displacement, x, of a mechanical system subjected to a time-dependent force, F by equation 1a. We also saw that the charge, Q, in an electrical circuit with a time-dependent potential difference across it could be represented by equation 1b. In both cases the variable, t,… Continue reading 24.5 Linear second-order systems
In post 24.3 we explored a model of how a population could control itself. This model was based on a differential equation – the most common way to describe how things change with time. This approach was used to describe exponential growth (post 18.15), waves (post 19.12), diffusion (posts 19.15 and 19.16), the behaviour of… Continue reading 24.4 Logistic difference equation: self-controlled growth and chaos
Before you read this, I suggest you read post 18.15. In post 22.17 we saw the effect of predators on the population of a species: the example I used was the natural control of a rabbit population by foxes. But, in post 16.5, we saw that a population will control itself, without the need for… Continue reading 24.3 Limits to growth: the logistic differential equation
In post 23.4, we saw that the interpretation of x-ray diffraction patterns from crystals involves a temperature factor because atoms are displaced from their ideal lattice points by thermal vibrations. We have seen that heat is kinetic energy of atomic molecular motion (post 16.35). In a crystalline solid, the atoms must be oscillating about their… Continue reading 24.2 Thermal vibration of an atom in a crystal: the temperature factor in x-ray crystallography
I am going to calculate the Fourier transform of a Gaussian function because I want to use the result in a later post. I can define the Gaussian function, g(x) as an exponential function of the form g(x) = exp(-ax2). Here, exp(y) is another way of writing ey where e a number with the approximate… Continue reading 24.1 Fourier transform of a Gaussian function
Before you read this, I suggest you read section 10 of post 22.10. In post 22.10 we saw that integration reverses the process of differentiation. Suppose f and f’ are functions of x only and that f’ = df/dx. (2) To evaluate the integral we recognise the function f that satisfies equation 2. Then Sometimes… Continue reading 23.11 Integration by parts
At the beginning of post 23.4, I described two routes to understanding x-ray crystallography. If you prefer the second, less mathematical, route – ignore this post! Before you read this, I suggest you read post 23.8. In post 22.14, we saw that the x-ray diffraction pattern of an object is given by the Fourier transform… Continue reading 23.10 Frequency analysis and x-ray crystallography
