Question

The potential energy for a conservative force system is given by $U=a{{x}^{2}}-bx$ , where a and b are constants find out
(1). The expression of force
(2). Equilibrium Position
(3). Potential energy at equilibrium

Answer 1

Hint: In a conservative force system if we are given potential energy as a function of x, where x is position then magnitude of force can be found by partial derivation of function of potential energy with respect to x. Equilibrium position can be found by equating force function with x to zero. This can also be done by finding the position of minima or maxima of potential energy function.

Complete Step By Step Solution:
Potential energy is negative of work done by conservative force
$U=-W$, where U is potential energy and W is work done by force
Work of a force is integration of force with respect to position.
$W=\int{F\cdot ds}$, where W is work done by force, F is force and $ds$ is displacement in very small time ($ds\to 0$ ).
Force can be archived by derivating Potential energy with respect to position and taking negative of it. Here,
$U=a{{x}^{2}}-bx$
$F=-\dfrac{dU}{dx}$
$F=b-2ax$, where F is force, a, b are constants.
Equilibrium position is the position where net force is 0. So here,
$F=b-2ax=0$
$x=\dfrac{b}{2a}$
Potential energy at equilibrium will be,
$U\left( \dfrac{b}{2a} \right)=a{{\left( \dfrac{b}{2a} \right)}^{2}}-b\left( \dfrac{b}{2a} \right)$
$U=-\dfrac{{{b}^{2}}}{4a}$

Note: One must take care of the negative sign while finding force from potential energy or vice versa. People often forget to put negative signs after derivation as well as integration. We should remember,
$F=-\dfrac{dU}{dx}$
$U=-\int{F\cdot dx}$
The concept of potential energy is useful only when all force acting on body is conservative. This is because work of conservative force is independent of the path. Gravitational force, electrostatic force, etc. are examples of such forces. If the work done by force depends on the path taken by body, method of potential energy should be avoided