In post 19.2, we saw that the ideal gas equation pV = nRT, provides a good model for the behaviour of most gases except at very high pressures. Here p represents the pressure of the gas, V its volume, n the number of moles and R is the ideal gas constant.
In post 18.11, we saw that the concept of a simple harmonic oscillator can explain the behaviour of many oscillating systems. The idea behind this concept is that the energy dissipated is negligible, so no energy need be supplied for the oscillation to continue. The behaviour of a mechanical oscillator confined to a line is then given by
d2x/dt2 = –ω2x (1)
Here x is the displacement of the oscillator at time t and ω is the angular frequency of the oscillation. More details are given in post 18.11; an equation of the form of equation 1 is derived from the simple harmonic oscillator concept in post 18.6; a simple explanation of ω is given in post 16.14.
The fundamental difference between equation 1 and the ideal gas equation is that it contains something that is differentiated (post 17.4). Equation 1 is an example of a differential equation; the equation in the picture at the beginning of this post is also an example. The properties of a gas do not change with time. But the behaviour of an oscillator changes with time and so is described by a differential equation.
From the analogy between electrical and mechanical systems (post 18.24) the current in a circuit when a capacitor discharges through an inductor, when there is no additional energy source and the resistance is negligible (so no energy is supplied or dissipated) is given by
d2Q/dt2 = –ω2Q (2)
where ω = (1/LC)1/2. Here C is the capacitance of the capacitor (post 18.19) and L is the inductance (post 18.21) of the inductor. Equation 2 represents the behaviour of an electrical simple harmonic oscillator.
When an object is bouncing on a spring, we can usually assume that k and m are constant; in an electrical circuit, C and L are often constant. Then the values of x, in equation 1, and Q, in equation 2, depend only on t; we say that x and Q are functions of t.
When k and m are constant, equation 1 is a linear differential equation and can be solved. Similarly, when C and L are constant, equation 2 is a linear differential equation. The differential equation in the picture, at the beginning of this blog, is linear when k ,M and m are constant.
Equation 2 has two trigonometric solutions
x = acos(ωt) (3)
x = asin(ωt) (4)
where a is the amplitude of oscillation, as shown in post 18.6. Both are valid mathematical solutions. We must choose the one that best fits the physical system we wish to model, as described in post 18.6.
If equations 3 and 4 are solutions of equation 1, then so is
x = pacos(ωt) + qasin(ωt) (5)
where p and q are any constants. You can verify this result for yourself, using the method of appendix 3 in post 18.6. When p and q are constants, equation 5 is called a linear combination of equation 3 and 4. In general, any linear combination of solutions of a differential equation is itself a solution.
When p = 1 and q = i, the square root of minus 1 (post 18.16)
x = acos(ωt) + iasin(ωt) = aeiωt, (6)
according to Euler’s relation (post 18.17), where e is defined in post 18.15. Once again, you can verify that aeiωt is a solution of equation 1 using the method of appendix 3 in post 18.6 and noting that d(ent)/dt = nent, where n is a constant (post 18.15).
Previously, we have seen differential equations used to describe growth and decay (post 18.15) and simple harmonic oscillators (posts18.6, 18.7, 18.8 and 18.11) and the time-dependence of the current in electrical circuits (posts 18.20 and 18.22). In future posts, I hope to show how they can be used to describe waves and diffusion.
18.17 Euler’s relation, oscillations and waves
18.16 The square root of minus 1 and complex numbers
18.15 More about exponential growth: the number e
18.14 Wave shapes – Fourier series
17.37 More about torque – cross products of vectors
17.36 More about work – line integrals
17.19 Calculating distances from speeds – integration
17.4 Displacement, velocity and acceleration
17.3 Three-dimensional vectors