Conservative Force and Potential Energy

A rubber ball is dropped from a height of $5 \mathrm{m}$ on a plane, where the acceleration due to gravity is not shown. On bouncing, it rises to $1.8 \mathrm{m}$. The ball loses its velocity on bouncing by a factor of $\frac{\alpha }{5}$. Write value of $\alpha$.

If potential energy function for the force between two atoms in a diatomic molecule is approximately given by $U\left(x\right)=\frac{\mathrm{a}}{x^8}-\frac{\mathrm{b}}{x^4}$, where $\mathrm{a}$ and $\mathrm{b}$ are constants in standard SI units and $x$ is in metres. Find the dissociation energy of the molecule (in $\mathrm{J}$). [Take $\mathrm{a}=4 \mathrm{J} {\mathrm{m}}^8$ and $\mathrm{b}=20 \mathrm{J} {\mathrm{m}}^4$]

Select the correct alternative: When a conservative force does positive work on a body, the potential energy of the body _____.

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 shape of the potential energy vs displacement curve for a spring is:

Which of the following is true for an Unstable equilibrium ?

A system in _____ equilibrium accelerates away from its equilibrium position if displaced even slightly.

A block of mass $M$ in the  Fig. moving on the frictionless horizontal surface collides with the spring of spring constant

$k$ and compresses it by length $L$. The maximum momentum of the block after collision is

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

A massless platform is kept on a light elastic spring, as shown in the figure. When particle of mass 0.1 kg is dropped on the pan from a height of 0.24 m, the particle sticks to the pan, and the spring is compressed by 0.01 m. From what height should the particle be dropped to cause a compression of 0.04 m?