Question

The potential energy of a conservative system is given by $U=A{x}^2-Bx$, where $x$ represents the position of the particle. $A$ and $B$ are constants. Then

• A. Force acting on the system will be $(B-2Ax)$
• B. At $x=\frac{B}{2A}$, potential energy will be $\frac{-B^2}{4A}$
• C. $\frac{d^{2}U}{d{x}^2}=\text{constant}$, i.e., system is in neutral equilibrium
• D. None

Answer 1

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

$\frac{d^{2}U}{d{x}^2}=\text{constant}$, i.e., system is in neutral equilibrium

$U=A{x}^2-Bx$
$F=-\frac{dU}{dx}=-(2Ax-B)=B-2Ax$

$\text{at}x=\frac{B}{2A},U=A\times \frac{B^2}{4A^2}-B\times \frac{B}{2A}=\frac{-B^2}{4A}$

$\frac{d^{2}U}{d{x}^2}=2A=\text{constant}$, i.e., system is in neutral equilibrium