Question

A particle located in one dimensional potential field has potential energy function $U(x) = \dfrac{a}{{{x^2}}} - \dfrac{b}{{{x^3}}}$ where $a$ and $b$ are positive constants. The position of equilibrium corresponds to$x = $?
$(A)\dfrac{{3a}}{b}$
$(B)\dfrac{{2b}}{{3a}}$
$(C)\dfrac{{2a}}{{3b}}$
$(D)\dfrac{{3b}}{{2a}}$

Answer 1

Hint:The net force acting on a system in equilibrium is zero. In order to find the position of equilibrium, we will put the value of the potential energy in the expression of the force. Then we will differentiate the above expression and arrive at the answer.

Complete step by step answer:
Potential energy is that energy which a body possesses due to its relative stationary position in space or electric charge. Potential energy can also be said to be that inherent energy of the body relative to its static position to the other objects. Potential energy is basically one of the two main types of energy, while the other one being kinetic energy. There are further two types of potential energy, which are elastic potential energy and gravitational potential energy.
We know that the force can be expressed as,
$F = \dfrac{{ - dU}}{{dx}}$
The potential energy in this case is in the x-direction.
On putting the value $U(x) = \dfrac{a}{{{x^2}}} - \dfrac{b}{{{x^3}}}$ in the above expression, we get,
$F = - \dfrac{d}{{dx}}\left( {\dfrac{a}{{{x^2}}} - \dfrac{b}{{{x^3}}}} \right)$
 On differentiating the above expression,
$F = \left(\dfrac{{ - 2a}}{{{x^3}}} - \dfrac{{( - 3)b}}{{{x^4}}}\right)$
$F = \dfrac{{3b}}{{{x^4}}} - \dfrac{{2a}}{{{x^3}}}........(1)$
We also know that for equilibrium, the net force acting is always equal to zero.
So, ${F_{net}} = 0$
On equating the above expression with equation (1), we get,
$\dfrac{{3b}}{{{x^4}}} - \dfrac{{2a}}{{{x^3}}} = 0$
On taking the LCM,
$\dfrac{{3b - 2ax}}{{{x^4}}} = 0$
$3b - 2ax = 0$
On taking the negative term on the other side,
$2ax = 3b$
$x = \dfrac{{3b}}{{2a}}$
So, the correct answer is $(D)\dfrac{{3b}}{{2a}}$.

Note:A particular system is considered to be in a state of equilibrium, if it does not tend to undergo any further change of its own. In simple words it can also be said that equilibrium is that condition of a system when neither its state of motion nor its internal energy changes with the passage of time. So, the net force acting on a system in equilibrium is equal to zero.