272
6 Oscillations
which is the equation for angular SHM with ω 2 =
2
3m
(k 1 + 4k 2 ).
T =
2π
ω
= 2π
3m
2(k 1 + 4k 2 )
6.39 U (x) =
a
x 2 −
b
x
Equilibrium position is obtained by minimizing the function U (x).
dU
dx
= −
2a
x 3 +
b
x 2 = 0
x = x 0 =
2a
b
Measuring distances from the equilibrium position and replacing x by x +
2a
b
F = −
dU
dx
=
2a
x 3 −
b
x 2
F =
2a
(x + 2a/b) 3 −
b
(x + 2a/b) 2
=
2a
(2a/b) 3
1 +
bx
2a
−3
−
b
(2a/b) 2
1 +
bx
2a
−2
Since the quantity bx/2a is assumed to be small, use binomial expansion
retaining terms up to linear in x.
F = −
b 4 x
8a 3
Acceleration a =
F
m
= −
b 4 x
8a 3 m
= −ω 2 x
where ω =
b 4
8a 3 m
T =
2π
ω
= 4π
2ma 2
b 4