A particle of mass `1.5` kg moves along x-axis in a conservative force field. Its potential energy is given by `V(x)=2x^(3)-9x^(2)+12x,` where all quantities are written in SI units. The plot of this potential energy is given below.
It is seen that the particle can be in stable equilibrium at a point on x-axis, x_(0). When it is displaced slightly from this equilibrium position, It executes SHM with time period T. What is the range of total mechanical energy of the particle for which its motion can be oscillatory about a point

Options

A. `Elt5J`
B. `Elt8J`
C. `Elt12 J`
D. `E lt 9 J`

Answer

A

Explanation

`V_(max)at(dv)/(dx)=0`
`6x^(2)-18x+12=0`
`x^(2)-3x+2=0`
`x=1 , 2`
`(d^(2)v)/(dx^(2))=2x-3`
`=-1atx=1Rightarrow maximum`
`=1at x=2Rightarrow minimum`
Motion can be oscillatory about x=2
`EltE_(max)`
`E_(max)=2xx1^(3)-9xx1^(2)+12=5j`