Potential Energy and Conservative Forces

If the potential energy corresponding to a force$\overrightarrow{ F}=x^2\hat{i}+y\hat{j}$ at point $A (1, 1)$ is $U_A=10 \mathrm{J}$, find the potential energy ($U_B$) at point $B$($2, 3$).

A particle in a certain conservative force field has a potential energy given by $U=\frac{20xy}{z}$. The force exerted on it is

A satellite is moving in a circular orbit around the earth. The total energy of the satellite is$E=-2\times {10}^5 \mathrm{J}$. The amount of energy to be imparted to the satellite to transfer it to a circular orbit where its potential energy is $U=-1\times {10}^5 \mathrm{J}$ is equal to _____$\times {10}^6 \mathrm{J}$

A conservative force of ($-3\hat{i}$) $\mathrm{N}$ acts on a body of mass $3 \mathrm{kg}$. The change in potential energy of the body due to the force, when it undergoes a displacement $\left(6\hat{i}\right)m$ is

 

A block of mass $m=2 kg$ is pulled by a force$F=40 N$ upwards through a height $h=2 m$. Find the work done on the block by the applied force $F$ and its weight $mg$. (Take, $g=10 m s^{-2}$) 

The potential energy is defined for a system whose elements interact via conservative force. A hypothetical system has potential energy corresponding to a conservative force is given as $U=7x^{2}y- 2x$. Then
The force on system corresponding to above energy function is
 

The variation of potential $U$ of a body moving along $X$ axis varies with the position $x$ as shown in the figure.The body is in equilibrium state in

A particle is released from rest at origin. It moves under the influence of potential field $U=x^2-3x$.
Its kinetic energy at $x=2$ is

If a particle moving in a circle of radius $r$ has potential energy $u=\frac{k}{r^2}$, then find the force on the particle.

A particle is moving in a conservative field and its potential energy is given as $U=(6x-8y) \mathrm{J}$, where $x$ and $y$ co-ordinates are in meters. Particle has initial velocity $\left(2\hat{\mathrm{i}}-3\hat{\mathrm{j}}\right) \mathrm{m} {\mathrm{s}}^{-1}$ at origin. At $t=1 \mathrm{s}$, the angel between velocity and acceleration of particle is ${\cos }^{-1}\left(\frac{M}{5\sqrt{2}}\right)$

(Given: mass of the particle is $2\mathrm{kg}$) The value of $M$ is _____ .

Is work done is equal to negative of change in potential energy.

A particle of mass $2\mathrm{kg}$ is attached with a spring of stiffness $72 \mathrm{N} {\mathrm{m}}^{-1}$. When block is at $x=1 \mathrm{m}$ it moves with maximum speed. The block moves with $24 \mathrm{m} {\mathrm{s}}^{-1}$ while passing through $x=4 \mathrm{m}$. Find out the $x$ coordinates in $\mathrm{m}$ where it's velocity will be zero.

The potential energy, $U$ in the joule of a particle of mass $2 \mathrm{kg}$ moving in $X – Y$ plane follows the law $U = 2x + 4y,$ where $(x, y)$ are the coordinates (in meter) of the particle. If the particle is at rest at,  $(4, 4)$ at time $t = 0$ then, [choose the incorrect statement]

A block of mass $3 \mathrm{kg}$ moving with a speed $1 \mathrm{m} {\mathrm{s}}^{-1}$ collides with a spring of spring constant $100 \mathrm{N} {\mathrm{m}}^{-1}$ as shown. Portion $AB$ of the surface is smooth while $BC$ is rough. The minimum value of the coefficient of friction for which block will not bounce back once the spring is compressed is $\frac{1}{n}$. Find the value of $n$(Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

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 $W_3$.

Which of the following forces is not conservative?

The $\mathrm{P}.\mathrm{E}.$ of a particle of mass $0.1 \mathrm{kg}$ moving along the $x$- axis is given by, $U=5x\left(x-4\right) \mathrm{J}$, where $x$ is in $\mathrm{metre}$. It can be concluded that the wrong option is

In a gravitational force field a particle is taken from $A$   to $B$ along different paths as shown in figure. Then