$A$ uniform chain of length $L$ and mass $M$ is placed on a smooth table such that one-fourth of its length hangs over the edge. Find the work done to pull the hanging part of the chain onto the table.

$A$ mass of $M \ kg$ is suspended by a weightless string of length $\ell$. The horizontal force required to displace it until the string makes an angle of $45^\circ$ with the initial vertical direction is:

$A$ uniform chain of length $2\,m$ is kept on a table such that a length of $60\,cm$ hangs freely from the edge of the table. The total mass of the chain is $4\,kg$. What is the work done in pulling the entire chain onto the table? (Take $g = 10\,m/s^2$) ................ $J$

$A$ number of identical cubical blocks of edge $a$ and mass $m$ are lying on a horizontal table. The work done on the blocks in arranging them in a column of height $(n + 1)a$ on the table is

The potential energy $(U)$ of a diatomic molecule is a function dependent on $r$ (interatomic distance) as $U = \frac{\alpha}{r^{10}} - \frac{\beta}{r^5} - 3$,where $\alpha$ and $\beta$ are positive constants. The equilibrium distance between two atoms will be $\left(\frac{2\alpha}{\beta}\right)^{\frac{a}{b}}$,where $a = \dots \dots \dots \dots$

If a body of mass $200\, g$ falls from a height $200\, m$ and its total $P.E.$ is converted into $K.E.$ at the point of contact of the body with the earth's surface,then what is the decrease in $P.E.$ of the body at the contact? $(g = 10\, m/s^2)$ ............ $J$