Question

Directions: The following question has four choices, out of which one or more are correct. The potential energy function of a particle moving in one dimension is U=k[x−x33a2], where a and k areconstants. Then,

• A. (A) equilibrium is stable at x = a/2
• B. (B) equilibrium is stable at x = -a
• C. (C) at stable equilibrium, PE = -2ka/3
• D. (D) equilibrium is stable at x = +a

Answer 1

Answer: (A), (D)

(A) equilibrium is stable at x = a/2

Answer 2

Explanation:
Force corresponding to potential energy is,F=−dUdx=−k[1−x2a2]For equilibrium, F=0. Therefore,1−x2a2=0x=±aFor stable equilibrium,d2Udx2>0d2Udx2=−2xa2d2Udx2|at x=a=−2a<0, so unstable d2Udx2|at x=−a=2a>0, so stable Potential energy at x=−a isU=k[−a+a33a2]=−2ka3Hence, this is the required solution.