The force acting on a body moving along x-axis varies with the position of the particle shown in the figure. The body is in stable equilibrium at
.

Options

A. `x = x _(1)`
B. `x = x _(2)`
C. both `x _(1) and x _(2)`
D. neither `x _(1) and x _(2)`

Answer

B

Explanation

Conservative force is equal to negative gradient of potential energy function, so `F =- (dU)/(dx)`
It means slope of potential energy curve is equal to negative of conservative force.
Now, for stable equilibrium
`(d ^(2) U )/( dx ^(2)) = (d)/(dx) ((dU )/(dx )) = ` Positive
i.e. About position of stable equilibruim the rate of change of slope of potential energy curve with respect to x must be positive. Now
(i) Just before `x_(2):`
Force is positive, so slope of potential energy curve is negative
(ii) At `x _(2):`
Force is zero so slope of potential energy curve is zero.
(iii) Just after `x_(2) :`
Force is negative so slope of potential energy curve is positive.
It means about `x_(2)` rate of change of slope of potential energy curve bis positive.