Physical Chemistry Third Edition

624 14 Classical Mechanics and the Old Quantum Theory

be analyzed more easily. Let the horizontal coordinatexof the mass equal zero at its

equilibrium position and assign it to be positive if the spring is stretched and negative

if the spring is compressed. If the magnitude ofxis not too large the force on the mass

due to the spring is given byHooke’s law:

Fx−kx (14.2-13)

wherekis called theforce constant. Hooke’s law fails if a spring is stretched beyond

its elastic limit or if it is sufficiently compressed that its coils touch each other.

Hooke’s law is named for Robert Hooke,
1635–1703, one of Newton’s
contemporaries and rivals.

The harmonic oscillator has the following properties: (1) thexcoordinate is not

limited to any finite range; (2) Hooke’s law is obeyed exactly for all values ofx; (3)

the mass of the spring is negligible; (4) there is no friction.

Newton’s second law, Eq. (14.2-7), provides theequation of motionof the har-

monic oscillator:

−kxm

d^2 x

dt^2

(14.2-14)

This is adifferential equation. It is calledlinearbecause the dependent variablexenters

only to the first power and is calledsecond orderbecause its highest-order derivative is

the second derivative. The solution to a differential equation is a function that gives the

dependent variable (xin this case) as a function of the independent variable (tin this

case). There can be more than one solution function for a given differential equation.

Thegeneral solutionof a differential equation is a family of functions that includes

nearly every solution of the equation.

Since both the sine and the cosine are proportional to the negative of their second

derivatives, a general solution for Eq. (14.2-14) can be written by inspection:

x(t)Asin(

√

k/mt)+Bcos(

√

k/mt) (14.2-15)

whereAandBare constants. This solution conforms to the fact that the general solution

of a linear second-order differential equation is represented by a formula that contains

two arbitrary constants. The expression for the velocity is obtained by differentiation:

vx(t)

dx

dt



√

k/m[Acos(

√

k/mt)−Bsin(

√

k/mt)] (14.2-16)

The solution to Eq. (14.2-14) can be obtained in a more systematic way. A general

linear differential equation with constant coefficients can be written

0 c 0 x+c 1

dx

dt

+c 2

d^2 x

dt^2

+ ··· (14.2-17)

where thec’s represent constants. It is a fact that a solution to a linear differential

equation with constant coefficients can be written as

x(t)eλt (14.2-18)

whereλis a parameter that is determined by the coefficients. This solution is called a

trial solutionbecause we try it in the equation. Substitution of this trial solution into

Eq. (14.2-14) gives

−keλtm

d^2 eλt

dt^2

mλ^2 eλt (14.2-19)