A particle of mass m is located in a one dimensional field where potential energy is given by:V (x) =A (1-cos px) where A and p are constants. Calculate the period of small oscillations of the particle?

Answer:

Given parameters:

A particle of mass m is located in a one-dimensional potential field and

Potential energy (P.E) = V(x)=A(1−cos px)

The force experienced by the particle of mass m is

F = – (dV/dx)

F = – ( d(A(1−cos px)/ dx)

F = – Apsinpx

For small oscillations

F = ≈−Ap²x

Acceleration (α) = F/m

α = – (Ap²/m)x

But α = F/m

α = -ω²x

So ω = √Ap²/m

The period of small oscillations of the particle (T) = 2π/ω

T = 2π/(√Ap²/m)

T = 2π√m/Ap²