System of Particles and Rotational Motion

The position vector of 1kg object is $\overrightarrow{r}=\left(3\widehat{i}-\widehat{j}\right)m$ and its velocity $\overrightarrow{v}=\left(3\widehat{j}+\widehat{k}\right)m{s}^{-1}.$ The magnitude of is angular momentum is $\sqrt[]{x}Nm$  where x is________.

Match List – I with List – II

Choose the correct answer from the options given below:

A shell of mass $m$ is at rest initially. It explodes into three fragments having mass in the ratio $2:2:1$ . If the fragments having equal mass fly off along mutually perpendicular directions with speed $v$ , the speed of the third (lighter) fragment is :

The angular speed of a fly wheel moving with uniform angular acceleration changes from 1200rpm to 3120 in 16 seconds. The angular acceleration in $\mathrm{r}\mathrm{a}\mathrm{d}/{\mathrm{s}}^2$  is:

The ratio of the radius of gyration of a thin uniform disc about an axis passing through its centre and normal to its plane to the radius of gyration of the disc about its diameter is :

Two objects of mass $10\mathrm{k}\mathrm{g}$ and $20\mathrm{k}\mathrm{g}$ respectively are connected to the two ends of a rigid rod of length 10 $\mathrm{m}$ with negligible mass. The distance of the centre of mass of the system from the $10\mathrm{k}\mathrm{g}$  mass is :

As shown in the figure, $a$ $$ bob of mass $m$ $$ is tied by $a$ $$ massless string whose other end portion is wound on a fly wheel (disc) of radius $r$ $$ and mass $$ $m$ . When released from rest the bob starts falling vertically. When it has covered a distance of $h$ $$ , the angular speed of the wheel will be

Three point particles of masses $10\mathrm{k}\mathrm{g},1.5\mathrm{k}\mathrm{g}$ $$ and $$ $2.5\mathrm{k}\mathrm{g}$ are placed at there corners of a right angle triangle of sides $4.0\mathrm{c}\mathrm{m},3.0;\mathrm{c}\mathrm{m}$ $$ and $$ $5.0\mathrm{c}\mathrm{m}$ as shown in the figure. The centre of mass of the system is at a point:

A cord is wound round the circumference of wheel of radius r. The axis of the wheel is horizontal and the moment of inertial about it is I. A weight mg is attached to the cord at the end. The weight falls from rest. After falling through a distance 'h', the square of angular velocity of wheel will be:

Four identical solid spheres each of mass 'm' and radius 'a' are placed with their centres on the four corners of a square of side 'b'. The moment of inertia of the system about side of square where the axis of rotation is parallel to the plane of the square is