Potential Energy & Equilibrium

Given below are two statements:

Statement I: An object moves from position $r_1$ to position $r_2$ under a conservative force field $\vec{F}$. The work done by the force is $W = -\int_{r_1}^{r_2} \vec{F} \cdot \overrightarrow{dr}$.

Statement II: Any object moving from one location to another location can follow infinite number of paths. Therefore, the amount of work done by the object changes with the path it follows for a conservative force.

In the light of the above statements, choose the correct answer from the options given below:

Potential energy ( $V$) versus distance ( $x$) is given by the graph. Rank various regions as per the magnitudes of the force ( $F$) acting on a particle from high to low.

Which one of the following forces cannot be expressed in terms of potential energy?

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason $\mathbf{R}$

Assertion A: In a central force field, the work done is independent of the path chosen. Reason R: Every force encountered in mechanics does not have an associated potential energy. In the light of the above statements, choose the most appropriate answer from the options given below

The potential energy function (in $J$) of a particle in a region of space is given as $U = \left(2x^2 + 3y^3 + 2z\right)$. Here $x, y$ and $z$ are in meter. The magnitude of $x$-component of force (in $N$) acting on the particle at point $P(1, 2, 3)\text{ m}$ is:

Potential energy as a function of r is given by $U = \frac{A}{r^{10}} - \frac{B}{r^5}$, where r is the interatomic distance, A and B are positive constants. The equilibrium distance between the two atoms will be:

Given below is the plot of a potential energy function U(x) for a system, in which a particle is in one dimensional motion, while a conservative force F(x) acts on it. Suppose that Emech = 8 J, the incorrect statement for this system is:

[ where K.E. = kinetic energy ]

The potential energy (U) of a diatomic molecule is a function dependent on r (interatomic distance) as

$U = \frac{\alpha}{r^{10}} - \frac{\beta}{r^5} - 3$

where, $\alpha$ and $\beta$ are positive constants. The equilibrium distance between two atoms will be $\left(\frac{2\alpha}{\beta}\right)^{\frac{a}{b}}$, where a = ___________.

If the potential energy between two molecules is given by

U = $-\frac{A}{r^6} + \frac{B}{r^{12}}$,

then at equilibrium, separation between molecules, and the potential energy are:

A particle with total energy E is moving in a potential energy region U(x). Motion of the particle is restricted to the region when

The potential energy of a particle in a force field is $U = \frac{A}{r^2} - \frac{B}{r}$ where A and B are positive constants and r is the distance of particle from the centre of the field. For stable equilibrium, the distance of the particle is

The potential energy of a system increases if work is done

A particle of mass m is released from rest and follows a parabolic path as shown. Assuming that the displacement of the mass from the origin is small, which graph correctly depicts the position of the particle as a function of time?

The potential energy function for the force between two atoms in a diatomic molecule is approximately given by $U(x) = \frac{a}{x^{12}} - \frac{b}{x^6},$ where $a$ and $b$ are constants and $x$ is the distance between the atoms. If the dissociation energy of the molecule is

$D = \left[U(x = \infty) - U_{at\ equilibrium}\right], \quad D$ is

The potential energy of a $1\text{ kg}$ particle free to move along the $x$-axis is given by $V(x) = \left(\frac{x^4}{4} - \frac{x^2}{2}\right)\text{ J}$. The total mechanical energy of the particle is $2\text{ J}.$ Then, the maximum speed (in $\text{m/s}$) is

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