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When a particle is mass m moves on the x...

When a particle is mass `m` moves on the `x-` axis in a potential of the from `V(x) = kx^(2)`, it performs simple harmonic motion. The corresponding thime periond is proportional to `sqrt((m)/(k))`, as can be seen easily asing dimensional analysis. However, the motion of a pariticle can be periodic even when its potential enem increases on both sides `x = 0` in a way different from `kx^(2)` and its total energy is such that the particel does not escape to infinity. consider a particle of mass `m` moving onthe `x-`axis . Its potential energy is `V(x) = omega (alpha gt 0`) for `|x|` near the origin and becomes a constant equal to `V_(0)` for `|x| ge X_(0)` (see figure)

If the total energy of the particle is `E`, it will perform is periodic motion why if :

A

`E lt 0`

B

`E gt0`

C

`V_(0) gt E gt 0`

D

`E gt V_(0)`

Text Solution

Verified by Experts

The correct Answer is:
C
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When a particle is mass m moves on the x- axis in a potential of the from V(x) = kx^(2) , it performs simple harmonic motion. The corresponding thime periond is proportional to sqrt((m)/(k)) , as can be seen easily asing dimensional analysis. However, the motion of a pariticle can be periodic even when its potential enem increases on both sides x = 0 in a way different from kx^(2) and its total energy is such that the particel does not escape to infinity. consider a particle of mass m moving onthe x- axis . Its potential energy is V(x) = alpha *x^4 (alpha gt 0 ) for |x| near the origin and becomes a constant equal to V_(0) for |x| ge X_(0) (see figure) If the total energy of the particle is E , it will perform is periodic motion why if :

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