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 $II$

$(A)$ $U _1( x )=\frac{ U _0}{2}\left[1-\left(\frac{ x }{ a }\right)^2\right]^2$ $(P)$ The force acting on the particle is zero at $x = a$.

$(B)$ $U _2( x )=\frac{ U _0}{2}\left(\frac{ x }{ a }\right)^2$ $(Q)$ The force acting on the particle is zero at $x=0$.

$(C)$ $U _3( x )=\frac{ U _0}{2}\left(\frac{ x }{ a }\right)^2 \exp \left[-\left(\frac{ x }{ a }\right)^2\right]$ $(R)$ The force acting on the particle is zero at $x =- a$.

$(D)$ $U _4( x )=\frac{ U _0}{2}\left[\frac{ x }{ a }-\frac{1}{3}\left(\frac{ x }{ a }\right)^3\right]$ $(S)$ 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 $x=-a$.

Q: 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 $II$ $(A)$ $U _1( x )=\frac{ U _0}{2}\left[1-\left(\frac{ x }{ a }\right)^2\right]^2$ $(P)$ The force acting on the particle is zero at $x = a$. $(B)$ $U _2( x )=\frac{ U _0}{2}\left(\frac{ x }{ a }\right)^2$ $(Q)$ The force acting on the particle is zero at $x=0$. $(C)$ $U _3( x )=\frac{ U _0}{2}\left(\frac{ x }{ a }\right)^2 \exp \left[-\left(\frac{ x }{ a }\right)^2\right]$ $(R)$ The force acting on the particle is zero at $x =- a$. $(D)$ $U _4( x )=\frac{ U _0}{2}\left[\frac{ x }{ a }-\frac{1}{3}\left(\frac{ x }{ a }\right)^3\right]$ $(S)$ 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 $x=-a$.

Second Method $(A)$ $\overrightarrow{ F }=-\frac{ dU }{ dx } \hat{ i }=-\frac{ U _0}{2} 2\left(1-\left(\frac{ x }{ a }\right)^2\right) \times\left[-2\left(\frac{ x }{ a }\right) \times \frac{1}{ a }\right] \hat{ i }=2 U _0\left[1-\left(\frac{ x }{ a }\right)^2\right]\left[\frac{ x }{ a ^2}\right] \hat{ i }$ If $x=0 \Rightarrow \vec{F}=\frac{ U _0}{2}[2(1) \times 0]=\overrightarrow{0}, U =\frac{ U _0}{2}$ If $x=a \Rightarrow \vec{F}=\overrightarrow{0}, \& U=0$ If $x=-a \Rightarrow \vec{F}=\overrightarrow{0}, \& U=0$ $(B)$ $\overrightarrow{ F }=-\frac{ U _0}{2} \times 2\left(\frac{ x }{ a }\right) \times \frac{1}{ a } \hat{ i }=-\frac{ U _0 x }{ a ^2} \hat{ i }$ If $x=0 \Rightarrow \overrightarrow{ F }=0$ and $U =0$ If $x = a \Rightarrow \overrightarrow{ F }=-\frac{ U _0}{ a } \hat{ i }$ and $U =\frac{ U _0}{2}$ If $x=-a \Rightarrow \vec{F}=+\frac{U_0}{a} \hat{i}$ and $U=\frac{U_0}{2}$ For $(C)$ and $(D)$, similarly we can solve

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 $II$
$(A)$ $U _1( x )=\frac{ U _0}{2}\left[1-\left(\frac{ x }{ a }\right)^2\right]^2$	$(P)$ The force acting on the particle is zero at $x = a$.
$(B)$ $U _2( x )=\frac{ U _0}{2}\left(\frac{ x }{ a }\right)^2$	$(Q)$ The force acting on the particle is zero at $x=0$.
$(C)$ $U _3( x )=\frac{ U _0}{2}\left(\frac{ x }{ a }\right)^2 \exp \left[-\left(\frac{ x }{ a }\right)^2\right]$	$(R)$ The force acting on the particle is zero at $x =- a$.
$(D)$ $U _4( x )=\frac{ U _0}{2}\left[\frac{ x }{ a }-\frac{1}{3}\left(\frac{ x }{ a }\right)^3\right]$	$(S)$ 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 $x=-a$.

IIT 2015, Diffcult

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Solution

Second Method

$(A)$ $\overrightarrow{ F }=-\frac{ dU }{ dx } \hat{ i }=-\frac{ U _0}{2} 2\left(1-\left(\frac{ x }{ a }\right)^2\right) \times\left[-2\left(\frac{ x }{ a }\right) \times \frac{1}{ a }\right] \hat{ i }=2 U _0\left[1-\left(\frac{ x }{ a }\right)^2\right]\left[\frac{ x }{ a ^2}\right] \hat{ i }$

If $x=0 \Rightarrow \vec{F}=\frac{ U _0}{2}[2(1) \times 0]=\overrightarrow{0}, U =\frac{ U _0}{2}$

If $x=a \Rightarrow \vec{F}=\overrightarrow{0}, \& U=0$

If $x=-a \Rightarrow \vec{F}=\overrightarrow{0}, \& U=0$

$(B)$ $\overrightarrow{ F }=-\frac{ U _0}{2} \times 2\left(\frac{ x }{ a }\right) \times \frac{1}{ a } \hat{ i }=-\frac{ U _0 x }{ a ^2} \hat{ i }$

If $x=0 \Rightarrow \overrightarrow{ F }=0$ and $U =0$

If $x = a \Rightarrow \overrightarrow{ F }=-\frac{ U _0}{ a } \hat{ i }$ and $U =\frac{ U _0}{2}$

If $x=-a \Rightarrow \vec{F}=+\frac{U_0}{a} \hat{i}$ and $U=\frac{U_0}{2}$

For $(C)$ and $(D)$, similarly we can solve

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