The Chemistry Maths Book, Second Edition

348 Chapter 12Second-order differential equations. Constant coefficients

Then

y(θ 1 + 1 π) 1 = 1 c

1

e

i

ω

θ

e

i

ωπ

1 + 1 c

2

e

−i

ω

θ

e

−i

ωπ

andy(θ 1 + 1 π) 1 = 1 y(θ)ifπω 1 = 12 πnfor integer n. Therefore,ω 1 = 12 nforn 1 = 1 0, ±1, ±2,1=

and

y(θ) 1 = 1 c

1

e

2 inθ

1 + 1 c

2

e

− 2 inθ

0 Exercise 21

12.5 The harmonic oscillator

The simple (linear) harmonic oscillator (Figure 12.1) consists of a body moving in a

straight line under the influence of a force

F 1 = 1 −kx (12.27)

whose magnitude is proportional to the displacement xof the body from the fixed

point O, the point of equilibrium, and whose direction is towards this point. The

quantity kis called the force constant and the negative sign ensures that the force acts

in the direction opposite to that of the displacement.

By Newton’s second law of motion, the acceleration experienced by the body is

given by

(12.28)

and simple harmonic motion is therefore described by the differential equation

(12.29)

A variety of physical systems can be modelled in terms of simple harmonic motion;

the oscillations of a spring balance (when equation (12.27) is called Hooke’s law), the

swings of a pendulum, the oscillations of a diving springboard and, important in

chemistry, the vibrations of the atoms in a molecule or crystal.

m

dx

dt

kx

2

2

=−

Fm

dx

dt

=

2

2

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.........................................................................

.........................

..........

...

.........

....

.........

...........

...........

.......

• •

o x

x

equil ibrium mass m

force F=−kx

Figure 12.1