Simple Nature - Light and Matter

Summary of Notation
k spring constant
m mass of the oscillator
b sets the amount of damp-
ing,F=−bv
T period
f frequency, 1/T
ω (Greek letter omega), an-
gular frequency, 2πf, often
referred to simply as “fre-
quency”
ωo frequency the oscillator
would have without damp-
ing,

√

k/m

ωf frequency of the free vi-

brations

c sets the time scale for the

exponential decay enve-

lopee−ctof the free vibra-

tions

Fm strength of the driving

force, which is assumed to

vary sinusoidally with fre-

quencyω

A amplitude of the steady-

state response

δ phase angle of the steady-

state response

quality factor,Q, is defined asQ=ωo/ 2 c, and in the limit of weak

damping, whereω≈ωo, this can be interpreted as the number of

cycles required for the mechanical energy to fall off by a factor of

e^2 π= 535.49...Using this new quantity, we can rewrite the equation

for the frequency of damped oscillations in the slightly more elegant

formωf=ωo

√

1 − 1 / 4 Q^2.

self-check H

What if we wanted to make a simpler definition ofQ, as the number of

oscillations required for the vibrations to die out completely, rather than

the number required for the energy to fall off by this obscure factor?.

Answer, p. 1055

A graph example 43

The damped motion in figure g hasQ≈4.5, giving

√

1 − 1 / 4 Q^2 ≈

0.99, as claimed at the end of the preceding subsection.

Exponential decay in a trumpet example 44

.The vibrations of the air column inside a trumpet have aQof

about 10. This means that even after the trumpet player stops

blowing, the note will keep sounding for a short time. If the player

suddenly stops blowing, how will the sound intensity 20 cycles

later compare with the sound intensity while she was still blowing?

.The trumpet’sQis 10, so after 10 cycles the energy will have

fallen off by a factor of 535. After another 10 cycles we lose an-

other factor of 535, so the sound intensity is reduced by a factor

of 535×535 = 2.9× 105.

The decay of a musical sound is part of what gives it its charac-

ter, and a good musical instrument should have the rightQ, but the

Qthat is considered desirable is different for different instruments.

A guitar is meant to keep on sounding for a long time after a string

has been plucked, and might have aQof 1000 or 10000. One of the

reasons why a cheap synthesizer sounds so bad is that the sound

suddenly cuts off after a key is released.

3.3.3 Driven motion

The driven case is extremely important in science, technology,

and engineering. We have an external driving forceF=Fmsinωt,

where the constantFmindicates the maximum strength of the force

in either direction. The equation of motion is now

[1] ma+bv+kx=Fmsinωt

[equation of motion for a driven oscillator].

After the driving force has been applied for a while, we expect that

the amplitude of the oscillations will approach some constant value.

This motion is known as thesteady state, and it’s the most inter-

esting thing to find out; as we’ll see later, the most general type of

motion is only a minor variation on the steady-state motion. For

180 Chapter 3 Conservation of Momentum