
204 Classical Mechanics
© 2010 Taylor & Francis Group, LLC
=
1
0
τ
τ
()d .
Because the motion of an oscillator is repetitive, the time average over one period is the same as
over many periods and is unique.
Now, for an oscillator whose motion obeys Equation 7.10, x = a sin(ω
0
t + θ
0
), and the period
τ=2π⁄ω
0
:
〈〉==
+
∫
∫
Tmxma
11
2
1
2
0
0
22
2
0
2
0
τ
ω
ωθ
π
τ
πω
cos( )
/
d
//ω
0
.
Because the integral is extended over a complete period, it does not matter what the value of the
initial phase θ
0
is, and we may conveniently set θ
0
= 0. Using the fact that
ω
π
ω
π
πω π
0
2
0
0
2
2
0
2
2
1
2
1
2
0
co
os
/
tt yd
we nd
Tm
a
11
2
1
4
2
0
0
τ
ω
τ
.
The potential energy is (again, θ
0
= 0)