A particle of unit mass is moving along the x-axis under the influence of a force and its total energy is conserved. Four possible forms of the potential energy of the particle are given in column I (a and U 0 are constants). Match the potential energies in column I to the corresponding statement(s) in column II. Column I Column II A. U 1 x = U 0 2 1 - x a 2 2 P. The force acting on the particle is zero at x = a B. U

Step-by-step solution

U 1 x = U 0 2 1 - x 2 a 2 2 F = - U 0 2 2 1 - x 2 a 2 - 2 x a 2 = 2 U 0 a 4 a 2 - x 2 x F = 2 U 0 a 4 x a - x a + x F = 0 , at x = 0 , a , - a x = - a , U = 0 , x = 0 , U = U 0 2 ⇒ Oscillate when total ME is less then U 0 2 A → P , Q , R , T B- U 0 x = U 0 2 x a 2 F = - U 0 x a 2 F = 0 , x = 0 B → Q , S C- U 3 x = U 0 2 x a 2 exp ⁡ - x 2 a 2 F = - U 0 2 2 x a 2 exp ⁡ - x 2 a 2 + x 2 a 2 exp ⁡ - x 2 a 2 - 2 x a 2 = - U 0 2 2 x a 4 exp ⁡ - x 2 a 2 a 2 - x 2 = U 0 a 4 e x 2 a 2 x x + a x - a x = 0 , x = - a , x = + a