A uniform rod $AB$ of mass $3m$ and length $4l,$ which is free to turn in a vertical plane about a smooth horizontal axis through $A,$ is released from rest when horizontal. When the rod first becomes vertical a point $C$ of the rod, where $AC=3l,$ strikes a fixed peg. The rod is brought to rest by the peg. The impulse exerted by the peg on the rod is $J$(in SI unit). The value of $\frac{3 J}{m\sqrt{3 gl}}$ is -

Four $2\mathrm{kg}$ masses are connected by $\frac{1}{4} m$ long spokes to an axle as in shown figure. A force $F$ of $24 N$ acts on a lever $\frac{1}{2} m$ long to produce an angular acceleration $\alpha .$ (in $\left.\mathrm{rad}{s}^{-2}\right)$ Determine the magnitude of $\alpha .$

A uniform metre scale of mass $m$ is suspended by two vertical string attached to its two ends as shown in figure. A body of mass $m$ is placed on the $80 \mathrm{cm}$ mark. The ratio of tension in string. is $\frac{p}{q}.$ Find the value of $q-p$ (where $p$ and $q$ are integers).

A tangential force $F$ acts at the top of a thin spherical shell of mass $m$ and radius $R.$ The acceleration of the shell if it rolls without slipping. is given by $\frac{wF}{m}.$.Find the value of $w.$

The pulley shown in fig. has a moment of inertia $I$ about its axis and its radius is $R.$ The magnitude of the acceleration of the two blocks is given by $\frac{(M-p\times m)g}{\left(M+m\right)+\frac{q\times I}{R^2}}.$ Then find the value of $p+q.$ (Assume that the string is light and does not slip on the pulley)

A turning moment of $100 \mathrm{Nm}$ is applied to a shaft which passes through a wheel of radius $R=50\mathrm{cm}.$ The force with which brakes must be applied to the wheel (see figure) so that the wheel does not rotate is...... $\times {10}^2 N.$ The coefficient of friction equals $0.25.$

A uniform rod of mass $8m$ and length $6a$ is lying on a horizontal table. Two point masses $m$ and $2m$ moving with speed $2v$ and $v$ respectively strike the rod and stick to it as shown in the figure. Kinetic energy of the system after collision is given by $\alpha \times m{v}^2.$ The value of $\alpha$ is

A stick of length $L$ and mass $M$ lies on a frictionless horizontal surface on which it is free to move in anyway. A ball of mass $m$ moving with speed $v$ as shown in fig. Mass of the ball so that it remains at rest immediately after collision is $\frac{M{L}^2}{\left({ L}^2+K{d}^2\right)}.$ The value of $K$ is.