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 $\mathrm{U}_{0}$ are constants). Match the potential energies in column I to the corresponding statement(s) in column II.

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

Accepted Answer

Force is . Equilibrium (F=0) occurs at extrema of U. (A) . Zero at x = 0, ±a. At x = -a, U = 0 (minimum, stable). Particle with E = U₀/4 can oscillate about x = -a. → (P)(Q)(R)(T) (B) . Zero at x = 0. Force always towards x = 0 (attractive) for |x| < a. → (Q)(S) (C) . Zero at x = 0, ±a. Force towar…

JEE Advanced · 2015 · 24 May 2015 · Paper 1 · Work, Power & Energy

Solution

More JEE Advanced questions on Work, Power & Energy

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 $x = - a$ .