The Chemistry Maths Book, Second Edition

350 Chapter 12Second-order differential equations. Constant coefficients

or, settingk 2 m 1 = 1 ω

2

, in the form of equation (12.20),

(12.35)

with general solution, in trigonometric form,

x(t) 1 = 1 d

1

1 cos 1 ωt 1 + 1 d

2

1 sin 1 ωt (12.36)

The state of the system is determined by the initial conditions. For example, let the

displacement and velocity at timet 1 = 10 be

x(0) 1 = 1 A, x′(0) 1 = 10 (12.37)

Then, for the initial displacement,

x(0) 1 = 1 A 1 = 1 d

1

1 cos 101 + 1 d

2

1 sin 101 = 1 d

1

For the velocity

x′(t) 1 = 1 −d

1

ω 1 sin 1 ωt 1 + 1 d

2

ω 1 cos 1 ωt

x′(0) 1 = 1 −d

1

ω 1 sin 101 + 1 d

2

ω 1 cos 101 = 1 d

2

ω 1 = 10

Therefored

1

1 = 1 A, d

2

1 = 10 , and the solution of the initial value problem is

x(t) 1 = 1 A 1 cos 1 ωt (12.38)

The graph of the solution is shown in Figure 12.3. The maximum displacement of the

body from equilibrium is the amplitudeA. The wavelength of the representative

curve is the periodτ 1 = 12 π 2 ω, the time taken for one complete oscillation. The inverse

of the period, ν 1 = 112 τ 1 = 1 ω 22 π, is called the frequencyof oscillation, the number of

oscillations in unit time. The quantityω 1 = 12 πνis called the angular frequency. The

frequency of oscillation is related to the force constant kbyω

2

1 = 1 k 2 m:

ων=, = (12.39)

k

m

k

m

1

2 π

dx

dt

x

2

+=ω 0

A

−A

0

t

x(t)

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π/ω 2 π/ω 3 π/ω

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τ=2π/ω

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Figure 12.3