Question

Potential energy function along x-axis in a certain force field is given as $U\left(x\right)=\frac{x^4}{4}-2x^3+\frac{11}{2}{x}^2-6x$
For the given force field

• A. the points of equilibrium are $x=1,x=2$ and $x=3$
• B. the point $x=2$ is point of unstable equilibrium
• C. the points $x=1$ and $x=3$ are points of stable equilibrium
• D. there exists no point of neutral equilibrium

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A the points of equilibrium are $x=1,x=2$ and $x=3$
B there exists no point of neutral equilibrium
C the point $x=2$ is point of unstable equilibrium
D the points $x=1$ and $x=3$ are points of stable equilibrium

For equilibrium points, $\frac{dU}{dx}=0$ $\Rightarrow$ $x^3-6x^2+11x-6=0$ $\Rightarrow$ $x=1,x=2$ and $x=3$

In stable equilibrium, $\frac{d^{2}u}{d{x}^2}>0$ $\Rightarrow$ $x=1$ and $x=3$ are stable

while $x=2$ is unstable

In neutral equilibrium, $\frac{dU}{dx}=0$ and $\frac{d^{2}u}{d{x}^2}=0$