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 :

A disc of mass $M$ and radius $R$ is free to rotate about its vertical axis as shown in the figure. A battery operated motor of negligible mass is fixed to this disc at a point on its circumference. Another disc of the same mass $M$ and radius $R / 2$ is fixed to the motor's thin shaft. Initially, both the discs are at rest. The motor is switched on so that the smaller disc rotates at a uniform angular speed $\omega$. If the angular speed at which the large disc rotates is $\omega / n$, then the value of $n$ is ___________.

A thin uniform rod of length $L$ and certain mass is kept on a frictionless horizontal table with a massless string of length $L$ fixed to one end (top view is shown in the figure). The other end of the string is pivoted to a point $\mathrm{O}$. If a horizontal impulse $P$ is imparted to the rod at a distance $x={L \over n}$ from the mid-point of the rod (see figure), then the rod and string revolve together around the point $\mathrm{O}$, with the rod remaining aligned with the string. In such a case, the value of $n$ is ___________.

A thin circular coin of mass $5 \mathrm{gm}$ and radius $4 / 3 \mathrm{~cm}$ is initially in a horizontal $x y$-plane. The coin is tossed vertically up ( $+z$ direction) by applying an impulse of $\sqrt{\frac{\pi}{2}} \times 10^{-2} \mathrm{~N}$-s at a distance $2 / 3 \mathrm{~cm}$ from its center. The coin spins about its diameter and moves along the $+z$ direction. By the time the coin reaches back to its initial position, it completes $n$ rotations. The value of $n$ is ________.

[Given: The acceleration due to gravity $g=10 \mathrm{~m} \mathrm{~s}^{-2}$ ]