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A particle of mass m is executing oscill...

A particle of mass m is executing oscillations about the origin on the x-axis , Its potential energy is U (x) = k [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

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

Text Solution

Verified by Experts

The correct Answer is:
A
`V = k|x|^(3)`
`F = -(dv)/(dx) = -3k |x|^(2)` …(1)
The equation of simple harmonic motion is given as
`x = a sin omega t`
`rArr m (d^(2)x)/(dt^(2)) = m (- a omega^(2) sin omega t) = - m omega^(2) x` ….(2)
Using (i) and (ii), we obtain
`3k |x|^(2) = m omega^(2) x`
`rArr omega = sqrt(2k x//m)`
`rArr T = 2pi sqrt((m)/(2kx)) rArr T = 2pi sqrt((m)/(3ka sin omega t))`
`rArr T prop (1)/(sqrta)`
Hence (a) is correct
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