MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
Physics 8.01
Table Problem W13D2-3: Small Oscillations Solution
A particle of effective mass m is acted on by a potential energy given by
U (x) = U 0 (!ax 2 + bx 4 )
where U 0 , a , and b are positive constants.
a) Find the points where the force on the particle is zero. Classify them as stable or unstable.
b) If the particle is given a small displacement from an equilibrium point, find the angular
frequency of small oscillation.
Solution:
The force on the particle is zero at the minimum of the potential energy:
0=
dU
= U 0 (!2ax + 4bx 3 ) .
dx
Thus the equilibrium points are x = 0 which is unstable, and
x=±
a
2b
which are stable equilibrium points. The second derivative of the potential energy is
given by
d 2U
= U 0 (!2a + 12bx 2 )
dx 2
If the particle is given a small displacement from x0 = + a / 2b then
d 2U
(x ) = 4aU 0 .
dx 2 0
So the angular frequency of small oscillations is given by
! 0 = 4aU 0 / m ,
and the period of small oscillations by
T = 2! / " 0 = 2! m / 4aU 0 .