Relation between Potential Energy and Conservative Force

A uniform chain of length $2 \mathrm{m}$ is kept on a table such that a length of $60 \mathrm{cm}$ hangs freely from the edge of the table. The total mass of the chain is $4 \mathrm{kg}$. What is the work done in pulling the entire chain in the table?

A particle is moved from $\left(0, 0\right)$ to $\left(a, a\right)$ under a force $F=(3\hat{\mathrm{i}}+4\hat{\mathrm{j}})$ from two paths. Path $1$ is $OP$ and path $2$ is $OQP$. Let $W_1$ and $W_2$ be the work done by this force in these two paths. Then,

If $W_1, W_2$ and $W_3$ represent the work done in moving a particle from $A$ to $B$ along three different paths $1, 2$ and $3$, respectively (as shown), in the gravitational field of a point mass $m$, find the correct relation between $W_1, W_2$ and $W_3$.

A spring when compressed by $4 \mathrm{cm}$ has $2 \mathrm{J}$ energy stored in it. The force required to extend it by $8 \mathrm{cm}$ will be

A long spring is stretched by x cm its PE is U. If the spring is stretched by Nx cm the PE stored in it will be -

The potential energy for a force field $\overrightarrow{\mathrm{F}}$ is given by $\mathrm{U}\left(\mathrm{x},\mathrm{ }\mathrm{y}\right)=\mathrm{ }\cos \left(\mathrm{x}+\mathrm{y}\right)$ . The force acting on a particle at position given by coordinates $\left(0,\mathrm{ }\frac{\mathrm{\pi }}{4}\right)$ is -

The potential energy of a particle of mass $5\mathrm{ }\mathrm{k}\mathrm{g}$ moving in $\mathrm{x}-\mathrm{y}$ plane is given by $\mathrm{U}=\left(-7\mathrm{x}+24\mathrm{y}\right)\mathrm{ }\mathrm{J}$ , where $\mathrm{x}$ and $\mathrm{y}$ are in meters. If particle starts from rest from origin, then speed of particle at $\mathrm{t}=2\mathrm{ }\mathrm{s}\mathrm{e}\mathrm{c}$ , is -

A block of mass $2\mathrm{ }\mathrm{k}\mathrm{g}$ is kept at origin at $\mathrm{t}=0$ and is having velocity $4\sqrt{5}\mathrm{m}/\mathrm{s}$ in positive $\mathrm{x}$ -direction. The only force on it is a conservative and its potential energy is defined as $\mathrm{U}=-\mathrm{ }{\mathrm{x}}^3+6{\mathrm{x}}^2+15$ $\left(\mathrm{S}\mathrm{I}\mathrm{ }\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{s}\right)$ . Its velocity when the force acting on it is minimum (after the time $\mathrm{t}=0$ ) is -

A Particle of mass $100\mathrm{ }\mathrm{g}\mathrm{m}$ moves in a potential well given by $\mathrm{U}=\mathrm{ }8{\mathrm{x}}^2-\mathrm{ }4\mathrm{x}+400$ Joule. Find its acceleration at a distance of $25\mathrm{ }\mathrm{c}\mathrm{m}$ from equilibrium in positive direction -

The potential energy of a particle of mass $5\mathrm{ }\mathrm{k}\mathrm{g}$ moving in the $\mathrm{x}\mathrm{y}$ plane is given as

$\mathrm{U}=-\mathrm{ }7\mathrm{x}\mathrm{ }+\mathrm{ }24\mathrm{ }\mathrm{y}\mathrm{ }\mathrm{J}\mathrm{o}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{s}$, $\mathrm{x}$ and $\mathrm{y}$ in meters. Initially at $\mathrm{t}=0$ the particle is at the origin $\left(0,\mathrm{ }0\right)$ moving with a velocity of $\mathrm{v}=6\left[2.4\mathrm{ }\widehat{i}\mathrm{ }+\mathrm{ }0.7\mathrm{ }\widehat{j}\mathrm{ }\mathrm{m}/\mathrm{s}\right]$, then -

The force acting on a body moving along $x-\text{axis}$ varies with the position of the particle as shown in the figure.



The body is in stable equilibrium at___

A man places a chain (of mass $\mathrm{m }$and lenght $\mathcal{l}$)on a table slowly. Initally the lower end of the chain just touches the table. The man drops the chain when half of the chain is in vertical position. Then work done by the man in this process is :

The potential energy of a particle of mass 5 kg moving in the $x-y$ plane is given by $U=(-7x\hat{i}+24y\hat{j})$ J, $x$ and $y$ being in metre. Initially at $t=0$ the particle is at the origin (0, 0) moving with a velocity of $(2.4\widehat{\mathbf{i}}+0.7\widehat{\mathbf{j}})\mathrm{m}\mathrm{s}{\mathrm{ }}^{-1}$. The magnitude of force on the particle is

The potential energy as a function of the force between two atoms in a diatomic molecules 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 position of stable equilibrium for the system of the two atoms is given

Among the following, the case in which the work done increases the potential energy is

A particle is moving under the influence of a force given by $F=kx$, where $k$ is a constant and $x$ is the distance moved. The energy (in $\mathrm{joule}$) gained by the particle is moving from $x=0$ to $x=3$ is

A spring of force constant $800 {\mathrm{N} \mathrm{m}}^{-1}$ has an extension of $5 \mathrm{cm}$. The work done in extending it from $5 \mathrm{cm}$ to $15 \mathrm{cm}$ is

Under the action of a force $F=Cx$, the position of a body changes from $0$ to $x.$The work done is

A force of $\left(5+3x\right) \mathrm{N}$ acting on a body of mass $20 \mathrm{kg}$ along the $x$-axis displaces it from $x=2 \mathrm{m}$ to $x=6 \mathrm{m}$. The Work done by the force is

A particle moves along the $x-$axis from $x=x_1 to x=x_2$ under the action of a force given by $F=2x$. Then the work done in the process is