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 2 x = U 0 2 x a 2 Q. The force acting on the particle is zero at x = 0 . C. U 3 x = U 0 2 x a 2 exp ⁡ - x a 2 R. The force acting on the particle is zero at x = - a D. U 4 x = U 0 2 x a - 1 3 x a 3 S. The particle experiences an attractive force towards x = 0 in the region x a T. The particle with total energy U 0 4 can oscillate about the point x = - a

Question

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 2 x = U 0 2 x a 2 Q. The force acting on the particle is zero at x = 0 . C. U 3 x = U 0 2 x a 2 exp ⁡ - x a 2 R. The force acting on the particle is zero at x = - a D. U 4 x = U 0 2 x a - 1 3 x a 3 S. The particle experiences an attractive force towards x = 0 in the region x a T. The particle with total energy U 0 4 can oscillate about the point x = - a, a-s,t;b-s;c-s,t;d-t;, a-s;b-r,s;c-r,s,t;d-s,t;, a-r,s;b-q;c-q,r;d-t;, a-p,q,r,t;b-q,s;c-p,q,r,s;d-p,r,t;

MARKS App · Accepted Answer

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 , F = 0 x = 0 , U = 0 even function hence minima C → P , Q , R , S D- U 4 x = U 0 2 x a - x 3 3 a 3 F = - U 0 a 1 a - x 2 a 3 = U 0 2 a 3 x 2 - a 2 x = + a , x = - a , F = 0 x = - a , U = - U 0 3 , x = + a , U = + U 0 3 Hence oscillate about x = - a if T . M . E U 0 3 D → P , R , T

	Column I		Column II
A.	$U_{1} (x) = \frac{U_{0}}{2} {[1 - {(\frac{x}{a})}^{2}]}^{2}$	P.	The force acting on the particle is zero at $x = a$
B.	$U_{2} (x) = \frac{U_{0}}{2} {(\frac{x}{a})}^{2}$	Q.	The force acting on the particle is zero at $x = 0$ .
C.	$U_{3} (x) = \frac{U_{0}}{2} {(\frac{x}{a})}^{2} \exp [- {(\frac{x}{a})}^{2}]$	R.	The force acting on the particle is zero at $x = - a$
D.	$U_{4} (x) = \frac{U_{0}}{2} [\frac{x}{a} - \frac{1}{3} {(\frac{x}{a})}^{3}]$	S.	The particle experiences an attractive force towards $x = 0$ in the region $\|x\| < a$
		T.	The particle with total energy $\frac{U_{0}}{4}$ can oscillate about the point $x = - a$

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