The Chemistry Maths Book, Second Edition

46 Chapter 2Algebraic functions

(2.24)

EXAMPLE 2.19Behaviour of a quadratic for large values of the variable

For values of|x|larger than about 100 the functionf(x) 1 = 1 x

2

1 + 1 x 1 − 11 differs fromx

2

by less than 1%. The difference decreases like 12 xas xincreases: 0.1% for|x| 1 = 110

3

,

0.001%, for|x| 1 = 110

5

,and 10

− 8

%for|x| 1 = 110

10

.

Quadratic functions are important in the physical sciences because they are used to

model vibrational motions of many kinds. The simplest kind of vibrational motion

is simple harmonic motion and, for example, a ball rolling forwards and backwards in

a parabolic container (a ‘parabolic potential well’) performs simple harmonic motion.

Other examples are the swings of a pendulum, the vibrations of atoms in molecules

and solids, the oscillating electric and magnetic fields in electromagnetic radiation.

EXAMPLE 2.20The classical simple harmonic oscillator

The simple (linear) harmonic oscillator consists of a body moving in a straight line

under the influence of a force

F 1 = 1 −kx

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

(positive) quantity kis called the force constant and the negative sign ensures that the

force acts in the direction opposite to that of the displacement. For a body of mass m,

the energy of the system is

Em kx=+

1

2

1

2

22

v

fx

x

a

x

a

x

a

x

ax

n

nn

n

()

=+ + ++ → →±

−− 12

2

0

as ∞

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

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

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Figure 2.9

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

•

O x

x

equi librium mass m

force F=−kx

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Figure 2.10

The Chemistry Maths Book, Second Edition

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