Equilibrium and Its Types

Q. The potential energy of a particle is given by, $U=2x^3-3x^2$. (in SI unit) The nature of equilibrium at $x=0$ & $x=1$ are

1. Stable, Unstable
2. Unstable, Stable
3. Unstable, Stable
4. Stable, Unstable

Q. The graph of potential energy ($U$) Vs position ($X$) is shown in the figure given below. Select the correct order regarding the equilibrium state of $\left(\text{a}\right),\left(\text{b}\right)$ & $\left(\text{c}\right)$.

1. Stable, Unstable, Neutral
2. Unstable, Stable, Neutral
3. Neutral, Stable, Unstable
4. Neutral, Unstable, Stable

Q. The potential energy variation between two atoms as a function of the distance between them is given by $U\left(x\right)=U_0[{\left(\frac{a}{x}\right)}^{12}-2{\left(\frac{a}{x}\right)}^6],U_0,a>0$.
Find the equilibrium position and state the nature of equilibrium (stable or unstable).

1. $x_0=2a$ and stable
2. $x_0=a$ and stable
3. $x_0=2a$ and unstable
4. $x_0=a$ and unstable

Q. A body is freely moving under the action of a conservative force. If the work done by this conservative force is 'a', change in kinetic energy 'b' & magnitude of change in potential energy is 'c', then

1. a = c
2. a + b=0
3. a + c = 0
4. b + c = 0

Q. The given plot shows the variation of the potential energy $\left(U\right)$ of interaction between two particles with respect to the distance $\left(r\right)$ separating them. Then,

1. $\text{B}$ and $\text{D}$ are equilibrium points
2. $\text{C}$ is a point of unstable equilibrium
3. The force of interaction between the two particles is attractive between points $\text{C}$ and $\text{D}$ and repulsive between $\text{D}$ and $\text{E}$
4. The force of interaction between the particles is repulsive between points $\text{E}$ and $\text{F}$.

Q. The potential energy of a particle is given by relation $U=x^3-6x^2$ (In SI units). The nature of equilibrium at $x=0$ is

1. Stable
2. Unstable
3. Neutral
4. Undefined

Q. Select the correct figure that represents the stable equilibrium point.

1. 
2. 
3. 
4. 

Q. The potential energy between two atoms in a molecule is given by $U=a{x}^3-b{x}^2$ where $a$ and $b$ are positive constants and $x$ is the distance between the atoms. The atom is in stable equilibrium when $x$ is equal to

1. $0$
2. $\frac{2b}{3a}$
3. $\frac{3a}{2b}$
4. $\frac{3b}{2a}$

Q. The potential energy function for the force between two atoms in a diatomic molecule is approximately given by $U\left(x\right)=\frac{a}{x^{12}}-\frac{b}{x^6}$, where $a$ and $b$ are constants and $x$ is the distance between the atoms. If the dissociation energy of the molecule is $D=[U_{x=\infty }-U_{\text{at equilibrium}}]$, $D$ is

1. $\frac{b^2}{2a}$
2. $\frac{b^2}{12a}$
3. $\frac{b^2}{4a}$
4. $\frac{b^2}{6a}$

Q. Select the correct figure(s) that represents unstable equilibrium point.

1. 
2. 
3. 
4. Both (b) & (c)

Q. The potential energy variation between two atoms as a function of the distance between them is given by $U\left(x\right)=U_0[{\left(\frac{a}{x}\right)}^{12}-2{\left(\frac{a}{x}\right)}^6],U_0,a>0$.
Find the equilibrium position and state the nature of equilibrium (stable or unstable).

1. $x_0=2a$ and stable
2. $x_0=a$ and stable
3. $x_0=2a$ and unstable
4. $x_0=a$ and unstable

Q.

A particle moves in a potential region given by $U=8x^2-4x+400J$. Its state of equilibrium will be

1. x = 25 m

2. x = 0.25 m

3. x = 0.025 m

4. x = 2.5 m

Q. The potential energy of a particle varies with $x$ (position) according to relation $U=27x^3-64x$. We get maximum speed at $x=$

1. $\frac{8}{9}$ units
2. $\frac{3}{4}$ units
3. $\frac{12}{7}$ units
4. $\frac{5}{9}$ units

Q. The potential energy of a particle in a force field is given by the equation $E=\frac{x^3}{3}-4x^2+15x-6$ Choose the correct statement among the options.

1. no points of equilibrium exist.
2. $x=3$ and $x=5$ are points of equilibrium.
3. There is a stable equilibrium at $x=3$ and unstable equilibrium at $x=5$.
4. There is neutral equilibrium at $x=3$ and $x=5$.

Q. The potential energy of a particle is given by relation $U=x^3-6x^2$ (In SI units). The nature of equilibrium at $x=0$ is

1. Stable
2. Unstable
3. Neutral
4. Undefined

Q. The potential energy of a particle moving along $x-$axis, under the action of conservative force, is given by $U=20+5\sin \left(4\pi x\right)$, where $U$ is in Joules and $x$ is in metres. Then which of the following is/are true?

1. at $x=\frac{7}{8}\text{m}$, particle is at equilibrium
2. at $x=\frac{7}{8}\text{m}$, particle is not at equilibrium.
3. at $x=\frac{3}{8}\text{m}$, particle is at equilibrium.
4. at $x=\frac{3}{8}\text{m}$, particle is not at equilibrium.

Q. The potential energy between two atoms in a molecule is given by $U=a{x}^3-b{x}^2$ where $a$ and $b$ are positive constants and $x$ is the distance between the atoms. The atom is in stable equilibrium when $x$ is equal to

1. $0$
2. $\frac{2b}{3a}$
3. $\frac{3a}{2b}$
4. $\frac{3b}{2a}$

Q. The potential energy of a particle varies with $x$ (position) according to relation $U=27x^3-64x$. We get maximum speed at $x=$

1. $\frac{8}{9}$ units
2. $\frac{3}{4}$ units
3. $\frac{12}{7}$ units
4. $\frac{5}{9}$ units