phy1020.DVI

5.2 The Vertical Spring

If a horizontal mass on a spring is turned to a vertical position, then the spring is stretched by an amount
x 0 Dmg=k, giving it a new equilibrium position. For the vertical spring, the potential energy is still given
byUD^12 kx^2 ,butxin this case refers to the distance from theoriginal(horizontal) equilibrium position.

5.3 Frequency and Period

The angular frequency!described earlier is a measure of how fast the oscillator oscillates; specifically, it
measures how many radians of its motion the oscillator moves through each second, where one complete
cycle of motion is2radians. A related quantity is thefrequencyf, which describes how many complete
cycles of motion the oscillator moves through per second. The two frequencies are related by

!D2f: (5.16)

You can think of!andf as really being the same thing, but measured in different units. The angular
frequency!is measured in units of radians per second (rad/s); the frequencyfis measured in units of hertz
(Hz), where 1 HzD1/sec.
The reciprocal of the frequency is theperiodT, and is the time required to complete one cycle of the
motion:

TD

1

f

D

2

!

: (5.17)

The period is measured in units of seconds. As shown in the plot ofx.t/(Fig. 5.1), the periodTis the time
between peaks in the motion.

5.4 Mass on a Spring

The discussion so far has applied to simple harmonic motion in general; there are many specific examples
of physical systems that act as simple harmonic oscillators. The most commonly cited example is a mass
mon a spring with spring constantk. The spring constantkis a measure of how stiff the spring is, and is
measured in units of newtons per meter (N/m). Specifically,kdescribes how much force the spring exerts
per unit distance it is extended or compressed.
A mass on a spring oscillates with angular frequency

!D

r

k

m

; (5.18)

and therefore has periodTD2=!,or

TD2

r

m

k

: (5.19)

It really doesn’t matter whether a mass on a spring moves horizontally on a frictionless surface, or bobs
up and down vertically. The motion is the same—the only difference is that if you take a horizontal spring
and hang it vertically, the equilibrium position will change because of gravity. The period and frequency of
motion will be the same.
The importance of the spring example is not that there are government laboratories filled with researchers
studying springs; rather the spring example serves as an important model and approximation for other prob-
lems. Often even a complicated force can beapproximatedas a linear force (Eq. (5.1)) over some limited