Two block $P$ of mass $20\mathrm{kg}$ and $Q$ of mass $10\mathrm{kg}$ are placed on a rough horizontal disk $\left(\mu =\frac{2}{3}\right)$ and are connected with a string which is along the diameter of disk as shown in the figure. Then (Here $C$ is centre of the disc)

At time $t=0$, a disk of radius $1 \mathrm{m}$ starts to roll without slipping on a horizontal plane with an angular acceleration of $\alpha =\frac{2}{3} \mathrm{rad} {\mathrm{s}}^{-2}$. A small stone is stuck to the disk. At $t=0$, it is at the contact point of the disk and the plane. Later, at time $t=\sqrt{\pi } \mathrm{s}$, the stone detaches itself and flies off tangentially from the disk. The maximum height (in $\mathrm{m}$) reached by the stone measured from the plane is $\frac{1}{2}+\frac{x}{10}$. The value of $x$ is [Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$.]

If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places.

A small roller of diameter $20 \mathrm{cm}$ has an axle of diameter $10 \mathrm{cm}$ (see figure below on the left). It is on a horizontal floor and a meter scale is positioned horizontally on its axle with one edge of the scale on top of the axle (see figure on the right). The scale is now pushed slowly on the axle so that it moves without slipping on the axle, and the roller starts rolling without slipping. After the roller has moved $50 \mathrm{cm},$ the position of the scale will look like (figures are schematic and not drawn to scale)

Two identical spherical balls of mass $M$ and radius $R$ each are stuck on two ends of a rod of length $2R$ and mass $M$(see figure). The moment of inertia of the system about the axis passing perpendicularly through the centre of the rod is

A solid cube of wood of side $2a$ and mass $M$ is resting on a horizontal surface as shown below.

The cube is free to rotate about the fixed axis $AB$. A bullet of mass $m\left(<<M\right)$ and speed $v$ is shot horizontally at the face opposite to $ABCD$ at a height $h$ above the surface to impart the cube an angular speed ${\omega }_c$, so that the cube just topples over. Then, ${\omega }_c$ is (Note the moment of inertia of the cube about an axis perpendicular to the face and passing through the center of mass is $\frac{2M{a}^3}{3}$)

A sphere of radius $a$ and mass $m$ rolls along a horizontal plane with constant speed $v_0$. It encounters an inclined plane at angle $\theta$ and climbs upward. Assuming that it rolls without slipping, how far up the sphere will travel?