Equilibrium and Its Types

Q. Potential energy of a system of particles is $U=\frac{\alpha }{3r^3}-\frac{\beta }{2r^2}$, where $r$ is distance between the particles. Here $\alpha$ and $\beta$ are positive constants. Which of the following are correct for the system?

1. Equilibrium separation between the particles is $\frac{\alpha }{\beta }$
2. For $r=\frac{\alpha }{\beta }$, the equilibrium is stable
3. For $r=\frac{\alpha }{\beta }$, the equilibrium is unstable
4. Work required to slowly move the particles to infinite separation from initial equilibrium position is $\frac{\beta }{\alpha }$

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. 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. Neutral, Unstable
4. Unstable, Neutral

Q. The potential energy between two atoms in a molecule is given by $U\left(x\right)=\frac{a}{x^{12}}-\frac{b}{x^6}$ where $a$ and $b$ are positive constants and $x$ is the distance between the atoms. The atom is in stable equilibrium when

1. $x={\left(\frac{2a}{b}\right)}^{\frac{1}{6}}$
2. $x={\left(\frac{11a}{5b}\right)}^{\frac{1}{6}}$
3. $x=0$
4. $x={\left(\frac{a}{2b}\right)}^{\frac{1}{6}}$

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 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. Potential energy of a particle moving along $x-$ axis is given by $U=(\frac{x^3}{3}-4x+6)$.
Here, $U$ is in joules and $x$ in metres. Find the position of stable and unstable equilibriums, respectively.

1. $x=2\text{m}$ and $x=-2\text{m}$
2. $x=-2\text{m}$ and $x=2\text{m}$
3. $x=0$ and $x=0$
4. $x=1\text{m}$ and $x=-1\text{m}$

Q. Two billiard balls $A$ and $B$, each of mass $50g$ and moving in opposite directions with speed of $5m/s$ each, collide and rebound with the same speed. If the collision lasts for ${10}^{–3}s,$ which of the following statements are true?

1. The impulse imparted to each ball is $0.25kgm/s$ and the force on each ball is $250N.$
2. The impulse and the force on each ball are equal in magnitude and opposite in direction.
   
3. The impulse imparted to each ball is $0.25kgm/s$ and the force exerted on each ball is $25\times {10}^{–5}N.$
4. The impulse imparted to each ball is $0.5Ns.$

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 function for the force between two atoms in a diatomic molecule which is approximately given as,
$U\left(x\right)=\frac{a}{x^{12}}-\frac{b}{x^6}$. Where $a\&b$ are constants and $x$ is the distance between the atoms. The disassociation energy of the molecule, $D=\left[U\right(x=\infty )-U_{equilibrium}]$ is

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

Q. Figure shows two identical particles 1 and 2, each of mass $m,$ moving in opposite directions with same speed $v$ along parallel lines. At a particular instant, ${\overrightarrow{r}}_1$ and ${\overrightarrow{r}}_2$ are their respective position vectors drawn from point $A$ which is in the plane of the parallel lines.


Choose the correct options:

1. Total angular momentum of the system about $A$ is
   
   $\overrightarrow{l}=|mv({\overrightarrow{r}}_1+{\overrightarrow{r}}_2)|⨀$
2. Total angular momentum of the system about $A$ is
   
   $\overrightarrow{l}=mv(d_2-d_1)⨂$
3. Angular momentum ${\overrightarrow{l}}_2$ of particle $2$ about $A$ is ${\overrightarrow{l}}_2=mv{r}_2⨀$
4. Angular momentum ${\overrightarrow{l}}_1$ of particle $1$ about $A$ is ${\overrightarrow{l}}_1=mv\left(d_1\right)⨀$

Q. A conservative force $\overrightarrow{F}$ is acting on a particle and the particle moves from Point C to D and then from D to C. The work done by the force is

1. W = 0
2. W > 0
3. $W\ne 0$
4. W < 0

Q. An object, initially at rest, explodes in three fragments. The momentum of two pieces is \(-3p\hat{i}\) and \(-4p\hat{j}\) where \(p\) is a positive number. The momentum of the third piece
\((i)\) will have magnitude \(5p\)
\((ii)\) will make an angle \(\tan^{-1}(4/3)\) with the \(x\)-axis
\((iii)\) will make an angle \(\tan^{-1}(3/4)\) with the \(x\)-axis
\((iv)\) will have magnitude \(7p\).

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 varies with distance x as shown in the graph.
The force acting on the particle is zero at

1. C
2. B
3. A and D
4. B and C

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. In head-on collision between two particles $A$ and $B$ of same mass, $A$ is moving with momentum $15kgm/s$, $B$ is stationary. During the impact, $B$ gives impulse $10N-s$ to $A$. The coefficient of restitution is

1. $\frac{1}{4}$
2. $1$
3. $\frac{1}{3}$
4. $\frac{1}{2}$

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

1. 
2. 
3. 
4. 

Q. A particle moving on $\text{x-axis}$ has potential energy, $u=(2-20x+5x^2)\text{J}$ along $\text{x-axis}$. The particle is released at $x=-3$.The maximum value of $x$ in $\text{m}$ will be

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. Assertion :The change in kinetic energy of a particle is equal to the work done on it by the net force. Reason: Change in kinetic energy of particle is equal to the work done only in case of a system of one particle.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. A set of $n$ identical cubical blocks lie at rest parallel to each other along a line on a smooth horizontal surface. The separation between the near surfaces of any two adjacent blocks is $L$. The block at one end is given a speed $v$ towards the next one at time $t=0$. All collisions are completely inelastic, then
(i) The last block starts moving at $t=\frac{(n-1)L}{v}$.
(ii) The last block starts moving at $t=\frac{n(n-1)L}{2v}$.
(iii) The center of mass of the system will have a final speed $v$.
(iv) The center of mass of the system will have a final speed $\frac{v}{n}$.

1. (i), (iv)
   
2. (ii), (iii)
   
3. (ii), (iv)
   
4. (i), (iii)
   

Q. Potential energy of a system of particles is $U=\frac{\alpha }{3r^3}-\frac{\beta }{2r^2}$, where $r$ is distance between the particles. Here $\alpha$ and $\beta$ are positive constants. Which of the following are correct for the system?

1. Equilibrium separation between the particles is $\frac{\alpha }{\beta }$
2. For $r=\frac{\alpha }{\beta }$, the equilibrium is unstable
3. For $r=\frac{\alpha }{\beta }$, the equilibrium is neutral.
4. Work required to slowly move the particles to infinite separation from initial equilibrium position is $\frac{\beta }{\alpha }$

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

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

Q. If work done by conservative force is X, change in potential energy is Y and change in kinetic energy during motion is Z, then which of the following is compulsorily true?

1. X - Y = 0
2. None of these
3. X + Y = 0
4. Y + Z = 0

Q. The potential energy between two atoms in a molecule is given by $U\left(x\right)=\frac{a}{x^{12}}-\frac{b}{x^6}$ where a and b are positive constants and x is the distance between the atoms. The atom is in stable equilibrium when

1. 
2. 
3. 
4. 

Q. A trolley runs from point P to Q along a track, as shown in the figure. At point Q, its potential energy is 50 kJ less than at point P. At point P, the trolley has kinetic energy 5 kJ. Between P and Q, the work done against friction is 10 kJ. What is the kinetic energy at point Q?

1. 35 kJ
2. 45 kJ
3. 55 kJ
4. 65 kJ

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 $x=5$.