Conservative Forces and Potential Energy

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

Assertion: Friction is a conservative force.
Reason: Friction does not depend upon mass of the body.

Frictional force is_______________.

Which of the following statement is not true?

The potential energy of a system increases if work is done

The potential energy of a particle is determined by the expression $U=\alpha \left(x^2+y^2\right)$, where $\alpha$ is a positive constant. The particle begins to move from a point with coordinates $\left(3, 3\right)$, only under the action of the potential field force. Then its kinetic energy $K$ at the instant when the particle is at a point with the coordinates $\left(1, 1\right)$ is

The potential energy of a particle under a conservative force is given by $U\left(x\right)=\left(x^2-3x\right) \mathrm{J}$. The equilibrium position of the particle is at:

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

Example of a non-conservative force is:

A spring $40 \mathrm{mm}$ long is stretched by the application of a force. If $10 \mathrm{N}$ force required to stretch the spring through $1 \mathrm{mm}$, then work done in stretching the spring through $40 \mathrm{mm}$ is

Potential energy curve (sinusoidal) is shown graphically for a particle. The potential energy does not depend on $y$ and $z$ coordinates. The maximum value of corresponding conservative force (in magnitude) for range $0<x<2$ is $n\pi$ (in Newton). Find the value of $n$.

When a block moves with some friction on ground,

The spring constant of the spring of a gun is $k$ newton/meter. The spring has been compressed $x$ meter from its normal position and a bullet of mass $m$ has been put in the barrel. If the friction is negligible then on firing the gun, the bullet shall have a velocity of $\left(\mathrm{m}/\sec \right):$

From a fixed support $\mathrm{O}$, hangs an elastic string of negligible mass and natural length $L$. Masses $m_1$ and $m_2$ attached at the lower end, elongate the string to $2 \mathrm{L}$. When $m_2$ is removed gently, $m_1$ is able to bounce back to $\mathrm{O}$. The ratio $m_2/m_1$ is (assume that the string does not obstruct the motion of $m_1$ )

Calculate the force $F\left(y\right)$ associated with the following one-dimensional potential energy :-

$U=a{y}^3-b{y}^2$