Question

A particle of mass (m) is executing oscillations about the origin on the (x) axis. Its potential energy is `V(x) = k|x|^3` where (k) is a positive constant. If the amplitude of oscillation is a, then its time period (T) is.
A. proportional to `(1)/(sqrta)`
B. independent of a
C. proportional to `sqrta`
D. proportaional to `a^(3//2)`

Answer 1

Correct Answer - A
`u=k|x|^(3)`
`U_("max")=ka^(3)`
The oscillation energy is given by `(1)/(2)ma^(2)omega^(2)`.
`therefore" "(1)/(2)ma^(2)omega^(2)=ka^(2)a`
or `omega^(2)=(2k)/(m)a`
or `omega=sqrt([((2k)/(m))a])`
`therefore" "T=(2pi)/(omega)=2pisqrt((m)/(2ka))`
`therefore" "Tprop(1)/(sqrta)`