When a stone is thrown vertically upwards and returns to the ground,at what point will its potential energy be maximum?

$A$ chain of mass $m$ and length $l$ is hanging freely from edge $A$ (as shown in diagram $I$). Calculate the work done to fold it as shown in diagram $(II)$.

$A$ 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).

While a person climbs stairs,the gravitational potential energy of the person increases. The source of this energy is

$A$ particle of mass $1 \ kg$ is free to move along the $x$-axis. Its potential energy is given by $U(x) = (\frac{x^2}{2} - x) \ J$. If the total mechanical energy of the particle is $2 \ J$,find the maximum speed of the particle.

If a body has a potential energy of $(4x^2 + 2x) \,J$ at a position $x = 2 \,m$, then the force acting on the body is: (in $\,N$)