A container of height 2 m , length 2 m and breadth 1 m is made of insulating vertical walls and two large area horizontal metal plates ( $\mathrm{M}_1$ and $\mathrm{M}_2$ ) which extend far beyond the vertical walls in all directions. The container is partitioned into two equal chambers with a thin insulating vertical wall. The partition wall contains a small hole of cross-sectional area $\sqrt{10} \mathrm{~cm}^2$ near its bottom edge. Initially the hole is closed and the left chamber of the container is completely filled with a liquid of dielectric constant $\epsilon_r=15$ and the right chamber is empty ( $\epsilon_r=1$ ). At time $t=0$, the hole is opened and the liquid flows from the left chamber to the right chamber. In both the chambers, the space above the liquid has $\epsilon_r=1$ and is maintained at atmospheric pressure. The schematic of the container at a time $t>0$ is shown in the figure.

[Given : acceleration due to gravity is $10 \mathrm{~ms}^{-2}$.]

The difference in the capacitance (in F) between the metal plates at $t=0$ and that at $t=500 \mathrm{~s}$ is $(8-n) \epsilon_0$, where $\epsilon_0$ is the permittivity of free space. The value of $n$ is :

A uniform circular disk of radius 0.2 m and mass 1 kg is pivoted at its top point $C$ such that it can rotate freely around $C$ in the $X Y$ plane, as shown in the figure. Initially, when the disk is at rest, a particle of mass 20 g , travelling along negative $x$ direction in the $X Y$ plane with speed $100 \mathrm{~ms}^{-1}$, hits the circumference of the disk at a point $P$. After collision the particle moves along negative $y$ direction at a speed of $90 \mathrm{~ms}^{-1}$.

[Given : the acceleration due to gravity $(\mathrm{g})=-10 \hat{\jmath} \mathrm{~ms}^{-2}$ ]

After the collision the disk starts to rotate around point $C$ in the $X Y$ plane. The maximum change in the height (in m ) of its center $O$ is :

A uniform circular disk of radius 0.2 m and mass 1 kg is pivoted at its top point $C$ such that it can rotate freely around $C$ in the $X Y$ plane, as shown in the figure. Initially, when the disk is at rest, a particle of mass 20 g , travelling along negative $x$ direction in the $X Y$ plane with speed $100 \mathrm{~ms}^{-1}$, hits the circumference of the disk at a point $P$. After collision the particle moves along negative $y$ direction at a speed of $90 \mathrm{~ms}^{-1}$.

[Given : the acceleration due to gravity $(\mathrm{g})=-10 \hat{\jmath} \mathrm{~ms}^{-2}$ ]

Amount of energy loss (in J ) in the collision is :