Work Power and Energy 5 Question 14
17. 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 statements in Column II
(2015 Adv.)
| Column I | Column II |
|---|---|
| A. $U _1(x)=\frac{U _0}{2} 1-\frac{x^{2}}{a}$ P. The force acting on the particle is zero at $x=a$ B. $U _2(x)=\frac{U _0}{2} \frac{x^{2}}{a}$ Q. The force acting on the particle is zero at $x=0$ C. $U _3(x)=\frac{U _0}{2} \frac{x^{2}}{a}{ }^{2} \exp -\frac{x^{2}}{a}$ R. The force acting on the particle is zero at $x=-a$ D. $U _4(x)=\frac{U _0}{2} \frac{x}{a}-\frac{1}{3} \frac{x^{3}}{a}$ 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 $x=-a$. |
Show Answer
Answer:
Correct Answer: 17. A-P, Q, R, T B-Q, S $\quad$ C-P, Q, R, S $\quad D-P, R, T$
Solution:
- (A) $F _x=\frac{-d U}{d x}=-\frac{2 U _0}{a^{3}}[x-a][x][x+a]$
$$ \begin{aligned} & F=0 \text { at } x=0, x=a, x=-a \\ & U=0 \text { at } x=-a \end{aligned} $$
$$ \text { and } \quad x=a $$
(B) $F _x=-\frac{d U}{d x}-U _0 \frac{x}{a}$
(C) $F _x=-\frac{d U}{d x}=U _0 \frac{e^{-x^{2} / x^{2}}}{a^{3}}[x][x-a][x+a]$
(D) $F _x=-\frac{d U}{d x}=-\frac{U _0}{2 a^{3}}[(x-a)(x+a)]$
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