A uniform, long thin chain of length $l$, with one end attached to a horizontal axle of cross-sectional radius $r$ $\left(\mathrm{r}<<l\right)$, is hung vertically down initially. The chain is then wound up slowly on the axle, doing an amount of work ${\mathrm{W}}_1$ . The chain is then cut into two equal parts (with length $\frac{\mathrm{l}}{2}$ each) and the two parts are wound one after the other on the same axle doing an amount of work ${\mathrm{W}}_2$ and ${\mathrm{W}}_3$ respectively. The ratio $\frac{{\mathrm{W}}_1}{{\mathrm{W}}_2+{\mathrm{W}}_3}$ is -

A mass of  $0.5 \mathrm{kg}$ moving with a speed of $1.5 \mathrm{m} {\mathrm{s}}^{-1}$ on a horizontal smooth surface, collides with a nearly weightless spring of force constant $\mathrm{k}=50 \mathrm{N} {\mathrm{m}}^{-1}$. The maximum compression of the spring would be

The graphs below show the magnitude of the force on a particle as it moves along the positive $X$-axis from the origin to $X=X_1$. The force is parallel to the $X$-axis and conservative. The maximum magnitude $F_1$ has the same value for all graphs. Rank the situations according to the change in the potential energy associated with the force, least (or most negative) to greatest (or most positive).

A large block of wood of mass $M=5.99 \mathrm{kg}$ is hanging from two long massless cords. A bullet of mass $m=10 \mathrm{g}$ is fired into the block and gets embedded in it. The (block $+$ bullet) then swing upwards, their center of mass rising a vertical distance $h=9.8 \mathrm{cm}$ before the (block $+$ bullet) pendulum comes momentarily to rest at the end of its arc. The speed of the bullet just before the collision is: (Take $g=9.8{ \mathrm{m} \mathrm{s}}^{-2}$)