If the potential energy of a particle of mass $0.1 \ kg$ moving along $x$-axis is $U(x) = 5x(x-4) \ J$,then the speed of the particle is maximum at a position of

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:

The potential energy of a particle oscillating along the $x-$axis is given by $U = 20 + (x - 2)^2$,where $U$ is in joules and $x$ is in meters. The total mechanical energy of the particle is $36\,J$. The maximum kinetic energy of the particle is ................ $J$.

$A$ particle with constant total energy $E$ moves in one dimension in a region where the potential energy is $U(x)$. The speed of the particle is zero where

The force acting on a particle is given by the formula $F(x) = -kx + ax^3$. Find the graph of its potential energy $U(x)$ for $x \ge 0$.

$A$ carpet of mass $M$ made of a material is rolled along its length in the form of a cylinder of radius $R$ and kept on a rough floor. If the carpet is unrolled without sliding to a radius $R/2$,the change in potential energy is ($g=$ acceleration due to gravity).