Physical Chemistry Third Edition

14.2 Classical Mechanics 623

on the floor. The floor, which hardly moves at all, throws you into the air because of

Newton’s third law.

If a force on a particle depends only on its position, the force can be derived from

apotential energy. For thezcomponent of such a force

Fz−

(

∂V

∂z

)

x,y

(14.2-11)

where the potential energy is denoted byV. Similar equations apply for thexandy

directions. See Appendix E for further information. The derivative in Eq. (14.2-11)

is apartial derivative, which means that if you have a formula representingV, you

differentiate with respect tozin the usual way, treating any other independent variables

(xandyin this case) as though they were constants.

The total energy of a particle subject to a potential energy is the sum of its kinetic

energyK and its potential energyV:

EK +V

1

2

mv^2 +V (14.2-12)

The total energy of an object subject to a potential energy obeys thelaw of conser-

vation of energy, which states that the total energy of the object is constant in time.

The Harmonic Oscillator

In order to show how Newton’s laws determine the behavior of a particle, we apply

them to a harmonic oscillator, which is a model system designed to represent a mass

attached to a stationary object by a spring, as shown in Figure 14.1. Amodel systemis

designed to imitate a real system, but is defined to have simpler properties so that it can

x 5 0x 5 x(t)

Stationary object

massm

x

Figure 14.1 A System Represented by a Harmonic Oscillator.