A particle of mass m is present in a region where the potential energy of the particle depends on the x-coordinate according to the expression `U=(a)/(x^2)-(b)/(x)`, where a and b are positive constant. The particle will perform.

Options

A. oscillatory motion but not simple harmonic motion about its mean position for small displacements
B. simple harmonic motion with time period `2pisqrt((8a^2m)/(b^4))` about its mean position for small displacements
C. neither simple harmonic motion nor oscillatory about its mean position for small displacements
D. none of the above

Answer

D

Explanation

`U=(a)/(x^2)-(b)/(x)`,`F=-(dU)/(dx)=-[(-2a)/(x^3)+(b)/(x^2)]`
At equilibrium position `F=0`
`x=x_0=(2a)/(b)`
At `x=x_0+trianglex`, i.e., at a displacement of `trianglex(lt lt ltx_0)` form mean position
`F=-[(bx-2a)/(x^3)]=-[(b(x_0+trianglex)-2a)/((x_0+trianglex)^3)]=-(btrianglex)/(x_0^3)`
`(b^2)/(8a^3)trianglex`
As `Fproptrianglex`, So particle performs simple harmonic motion with time period
`T=2pisqrt((8a^3m)/(b^4))`