Q)
(chatquestion)...A particle oscillates with amplitude A in a one-dimensional potential U (x) that is
symmetric about x = 0, i.e. U (x) = U(-x). (a) Show, from energy considerations,
that the velocity v of the particle at displacement x from the equilibrium position
(x = 0), is given by
v = /2[U (A) – U(x)]/m.
(b) Hence show that the period of oscillation T is given by
dx
T = 4,
V[I- U(x)/U(A)]
2U (A)
Simple Harmonic Motion
(c) If the potential U (x) is given by
U (x) = ax"
where a is a constant and n = 2, 4, 6, ..., obtain the dependence of the period T on
the amplitude A for different values of n = 2, 4, .... (Hint: Introduce the new variable
of integration = x/A in the above expression for the period T.)
...
