Physical Chemistry Third Edition

14.2 Classical Mechanics 627

Since a force is given as a derivative of a potential energy, a constant can be added to

a potential energy without changing the force or any other physical property.

A time-independent force is equal to the negative derivative of a potential energy.

The potential energy corresponding to the force of Eq. (14.2-13) is

V(x)

1

2

kx^2 +V 0 (14.2-32)

whereV 0 is a constant. Since the derivative of a constant is zero, we can always add

any constant to a potential energy without any physical effect. We choose to setV 0

equal to zero. Figure 14.3a shows the potential energy for the harmonic oscillator as a

function ofx, and Figure 14.3b shows the force due to this potential energy. For our

initial conditions, the potential energy is given as a function of time by

V 

k

2

x^20 cos^2

(√

k

m

t

)

(14.2-33)

The total energy,E, is the sum of the kinetic and potential energies:

EK +V 

1

2

kx^20

[

sin^2

(√

k

m

t

)

+cos^2

(√

k

m

t

)]



1

2

kx^20 (14.2-34)

The final equality follows from the trigonometric identity:

sin^2 (α)+cos^2 (α) 1 (14.2-35)

The total energy does not change during the oscillation, corresponding toconservation

of energy. A conserved quantity is one that remains constant, and is called aconstant

of the motion. The total energy of the harmonic oscillator is a constant of the motion.

–3

V(

x)

Fx

/k

0

z

(a)

x

(b)

1

2

3

4

–2 –1 0 1 2 3

–3

–3

–1

0

1

3

2

–2

–2 –1 0 1 2 3

Figure 14.3 Mechanical Variables

of a Harmonic Oscillator. (a) The

potential energy. (b) The force on the

oscillator.

The largest value ofxis called theturning point, because this is the point at which

the oscillator changes its direction of motion. At the turning point the kinetic energy

vanishes and the total energy equals the potential energy. If we denote the value ofx

at the turning point byxt,

EV(xt)

1

2

kx^2 t (14.2-36)

Therefore

xt

√

2 E

k

(14.2-37)

For our initial conditions,xtx 0 , so that

E

1

2

kx^20 (14.2-38)

as in Eq. (14.2-34).

A different version of the harmonic oscillator can be used as a model for a vibrating

diatomic molecule. This model oscillator consists of two movable objects connected

by a spring, as depicted in Figure 14.4. As shown in Appendix E our formulas hold for

this model if we letxrepresent the distance between the two objects and replace the

massmby thereduced massμ:

μ

m 1 m 2

m 1 +m 2

(definition of the reduced mass) (14.2-39)