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A particle of mass (m) is executing osci...

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 `1sqrt(a)`

B

independent of `a`

C

proportional to `sqrt(a)`

D

proportional to `a^(3//2)`

Text Solution

Verified by Experts

`V=k|x|^(3)`
`F=(dv)/(dx)=-3k|x|^(2)`…….`(1)`
The equation of simple harmonic motion is given as
`x=a sinomega t`
`rArrm(d^(2)x)/(dt^(2))=m(-aomega^(2)sinomegat)=-momega^(2).x`……`(2)`
Using `(1)` and `(2)`, we obtain
`3k|x|^(2)=momega^(2)xrArromega=sqrt(3kx//m)`
`rArr T=2pisqrt((m)/(3kx))rArrT=2pisqrt((m)/(3kasinomegat))`
`T prop (1)/(sqrt(a))`
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