Question

In the figure shown there is the plot of a potential energy function of a conservative system $U$ versus $x.$ Which of the following statements is correct ?

• A. The force has the greatest magnitude at point $E_{2}.$
• B. The force is positive and greater than zero at points $E_{1},E_{2},E_{3}$ and $E_{4}.$
• C. The system is in stable equilibrium
  at $E_{3}.$
• D. The system is in neutral equilibrium
  at $E_{1}.$

Answer 1

Answer: (D)

$F=-\frac{\partial U}{\partial x}$
Force $F$ versus x is drawn in Figure. At points $E_{1},E_{2},E_{3}$ and $E_{4}$
$F=0$
$F$ has the greatest magnitude nearer to point $E_{1}.$
If $U$ is minimum i.e., $\frac{\partial U}{\partial x}=0$ and $\frac{\partial^{2} U}{\partial x^{2}}=$ positive, equilibrium is stable.
If $U$ is maximum i.e., $\frac{\partial U}{\partial x}=0$ and $\frac{\partial^{2} U}{\partial x^{2}}=$ negative, equilibrium is unstable.
If $U$ is constant i.e., $\frac{\partial U}{\partial x}=0$ and $\frac{\partial^{2} U}{\partial x^{2}}=0$ equilibrium is neutral.