A body of mass m is situated in potential field `U(x)=U_(o)(1-cospropx)` when, `U_(o)` and `prop` are constants. Find the time period of small oscillations.

Explanation

Given potential energy associated with the field
`U(x)=U_(0)(1-cospropx)" ".....(i)`
Now, force `F=-(dU(x))/(dx)`
`:.["for conservatine force f, we can write f"=(-du)/(dx)]`
[We have assumed the field the to be conservative]
`F=-(d)/(dx)(U_(0)-U_(0)cospropx)=-U_(0)propsinpropx`
`F=-U_(0)prop^(2)x" " .....(ii)`
[ `:.` for small oscillations propx is small,sinpropx`~~`propx]
`rArrFprop(-x)`
As, `U_(0),prop` being constant.
`:.` Motion is SHM for small oscillations. `F=-momega^(2)x" " .....(ii)`
Comparing Eqs. (iii) and (iii),we get
`momega^(2)=U_(0)prop^(2)`
`momega^(2)=U_(0)prop^(2)" or " omega=sqrt(U_(0prop^(2))/(m)`
Time period `T=(2pi)/(omega)2pisqrt((m)/(U_(0)prop^(2)))`
Not The motion is SHM only for small oscillations and hence, the time period is valid only in case of small oscillations.