A thin circular coin of mass $5\,\text{gm}$ and radius $4/3\,\text{cm}$ is initially in a horizontal $xy$-plane. The coin is tossed vertically upward (in $+z$ direction) by applying an impulse of $\sqrt{\frac{\pi}{2}}\times 10^{-2}\,\text{N-s}$ at a distance $2/3\,\text{cm}$ 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 $g = 10\,\text{m s}^{-2}$]

Step 1: Relate linear impulse to initial vertical velocity.

The linear impulse $J$ determines the initial vertical velocity $v$.

$$J = mv$$ $$v = \frac{J}{m}$$

Step 2: Relate angular impulse to angular velocity.

The angular impulse about the diameter is equal to the change in angular momentum.

$$Jd = I\omega$$ $$\omega = \frac{Jd}{I}$$

Step 3: Determine the moment of inertia.

The moment of inertia of a thin circular coin of mass $m$ and radius $R$ about its diameter is used to find the angular velocity.

$$I = \frac{1}{4}mR^2$$ $$\omega = \frac{4Jd}{mR^2}$$

Step 4: Calculate the time of flight.

The time $t$ for the coin to return to its initial position is given by the vertical motion under gravity.

$$t = \frac{2v}{g}$$ $$t = \frac{2J}{mg}$$

Step 5: Express the number of rotations.

The number of rotations $n$ is the total angle rotated divided by $2\pi$.

$$n = \frac{\omega t}{2\pi}$$ $$n = \frac{4J^2 d}{\pi m^2 R^2 g}$$

Step 6: Substitute the given values to calculate $n$.

Substitute $m = 5 \times 10^{-3}\text{ kg}$, $R = \frac{4}{3} \times 10^{-2}\text{ m}$, $J = \sqrt{\frac{\pi}{2}} \times 10^{-2}\text{ N}\cdot\text{s}$, $d = \frac{2}{3} \times 10^{-2}\text{ m}$, and $g = 10\text{ m/s}^2$.

$$n = \frac{4 \left(\sqrt{\frac{\pi}{2}} \times 10^{-2}\right)^2 \left(\frac{2}{3} \times 10^{-2}\right)}{\pi \left(5 \times 10^{-3}\right)^2 \left(\frac{4}{3} \times 10^{-2}\right)^2 (10)}$$ $$n = \frac{4 \left(\frac{\pi}{2} \times 10^{-4}\right) \left(\frac{2}{3} \times 10^{-2}\right)}{\pi \left(25 \times 10^{-6}\right) \left(\frac{16}{9} \times 10^{-4}\right) (10)}$$ $$n = 30$$

Answer: 30