Physical Chemistry Third Edition

1268 E Classical Mechanics

If the forces on the particles of a system depend only on the particles’ positions,

these forces can be derived from a potential energy. Consider motion in thezdirection.

Ifz′andz′′are two values ofz, the difference in the potential energyVbetween these

two locations is defined to equal the reversible work done on the system by an external

agent to move the particle fromz′toz′′.

∆V V(z′′)−V(z′)

∫z 2

z 1

Fext(rev)(z)dz (E-3)

Since only the difference in potential energy is defined in Eq. (E-3), we have the option

of deciding at what state of the system we want to set the potential energy equal zero.

We accomplish this by adding an appropriate constant to a formula for the potential

energy.

The external force Fext(rev)must exactly balance the force due to the other particles

in order for the process to be reversible:

Fext(rev)(z)−Fz (E-4)

By the principles of calculus, the integrand in Eq. (E-3) is equal to the derivative of the

functionV, so that

Fz−

dV

dz

(E-5)

In the case of motion in three dimensions, analogous equations for thexandycompo-

nents can be written, and the vector force is given by

FiFx+jFy+kFz−i

∂V

∂x

−j

∂V

∂y

−k

∂V

∂z

−∇V (E-6)

where the symbolsi,j, andkstand for unit vectors in thex,y, andzdirections, respec-

tively, and where the symbol∇(“del”) stands for the three-termgradient operator

expressed in the right-hand side of the first line of Eq. (E-6). A system in which no

forces occur except those derivable from a potential energy is called aconservative

system. It is a theorem of mechanics that the energy of such a system is constant, or

conserved.

E.2 Derivation of the Wave Equation for

a Flexible String

The wave equation for the flexible string was stated in Section 14.3. We now derive this

equation. Consider a small portion of the string lying betweenxandx+∆x, as shown

in Figure E.1. The tension force on the left end of the string segment is denoted byF 1 ,

and the force on the right end is denoted byF 2. Since the string is perfectly flexible,

no force can be put on the string by bending it. The force exerted on one portion of the

string by an adjacent portion is tangent to the string at the point dividing the portions,

and has a magnitude equal toT, the tension force at each end of the string. If the string

is straight, the forces at the ends of a given segment will cancel. If the string is curved,

the forces at the two ends of a portion of the string will not cancel.

Consider the segment of the string lying betweenxx′andxx+dx. We denote

the angles between thexaxis and the two tangent lines at the ends of the segment by