Physical Chemistry , 1st ed.

absolute position of the mass at the starting time, when t0). We get this
equation by solving the appropriate equations of motion, whether expressed in
Newton’s or Lagrange’s or Hamilton’s format.
It takes a certain time, seconds, for the oscillator to complete one full
cycle. Therefore, in 1 second, there will be 1/ oscillations. In a sinusoidal
motion, one cycle corresponds to an angular change of 2. The frequencyof
the oscillator in number of cycles per second or simply 1/second (s^1 ; another
SI-approved name for s^1 is hertz,or Hz) is defined as (Greek nu) and is
equal to



1



2

1





m

k

 (11.3)

The frequency is independent of the displacement. Such relationships have
been known since the late 1600s. Familiar harmonic oscillators include masses
on springs and clock pendulums.

Example 11.1

Assuming units of N/m for the force constant and kg for mass, verify that

equation 11.3 yields units of s^1 for frequency.

Solution

Recall that newton is a composite unit and that

1 N  1 

kg

s

2

m



The basic units for kare therefore



kg

s

2

m

 m 

k

m

g

s

m

 2 

k

s^2

g



Since the 1/2term doesn’t have any units associated with it, the units from

equation 11.3 become



kg

k

/

g

s^2



s

1

^2 

1

s

s^1

thus confirming that the frequency has units of s^1.

Example 11.2

a.For small displacements, a clock’s pendulum can be treated as a harmonic

oscillator. A pendulum has a frequency of 1.00 s^1. If the mass of the pen-

dulum is 5.00 kg, what is the force constant acting on the pendulum in units

of N/m? What is this force constant in units of mdyn/Å?

b.Calculate the similar force constant for a hydrogen atom having mass

1.673 

10 ^27 kg, attached to an atomically flat metal surface and vibrating

with a frequency of 6.000 

1013 s^1.

Solution

a.One need simply substitute into equation 11.3. Using units consistent with

N/m for the force constant, the equation looks like this:

1.00 s^1 

2

1





5.00

k

kg



11.2 The Classical Harmonic Oscillator 317