Question

A body of mass m is situated in a potential field `U(x)=U_(0)(1-cosalphax)` when `U_(0)` and `alpha` are constant. Find the time period of small oscialltions.

Answer 1

`U_((x))=U_(0)(1-cosalphax)`
`F=-(d(U_(X)))/(dx)=-(d)/(dx)[U_(0)(1-cosalphax)]=-[0+U_(0)(sinalphax)xxalpha]=-U_(0)alphasinalphax`
Since `alphax` is small, so `alphax~~alphax`
`F=-U_(0)alphaxxalphax=-U_(0)alpha^(2)x` …(i)
From (i) , `Fpropx` and `-ve` sign shows that F is directed towards mean position, hence body if left free will execute linear SHM. In this case,
Spring factor `=U_(0)alpha^(2)`
Inertia factor `=` mass of body `=`m
`:.` Time period `T=2pisqrt((i n ertia fact o r)/(spri ng fact o r))=2pisqrt((m)/(U_(0)alpha^(2)))`