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 $\frac{R}{2}$ 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 $\frac{\omega}{n}$, then the value of $n$ is ______.

Step 1: Identify the parameters of the system.

The system consists of a large disc of mass $M$ and radius $R$, and a small disc of mass $M$ and radius $\frac{R}{2}$ attached to a motor on the circumference of the large disc.

Step 2: Apply conservation of angular momentum.

Since there are no external torques acting on the system about the vertical axis of rotation, the total angular momentum is conserved.

$$L_{\text{initial}} = L_{\text{final}} = 0$$

Step 3: Determine the moments of inertia.

Calculate the moment of inertia of the large disc about its central axis, and the small disc about its own axis.

$$I_{\text{big}} = \frac{1}{2}MR^2$$ $$I_{\text{small}} = \frac{1}{2}M\left(\frac{R}{2}\right)^2 = \frac{MR^2}{8}$$

Step 4: Express the components of the final angular momentum.

The final angular momentum consists of the spin of the large disc, the spin of the small disc, and the orbital motion of the small disc's center of mass at a distance $R$ from the main axis.

$$L_{\text{big}} = I_{\text{big}}\Omega$$ $$L_{\text{spin}} = I_{\text{small}}\omega$$ $$L_{\text{orb}} = MR^2\Omega$$

Step 5: Set up the conservation equation.

Sum the individual angular momentum components to zero.

$$L_{\text{big}} + L_{\text{spin}} + L_{\text{orb}} = 0$$ $$\frac{1}{2}MR^2\Omega + \frac{MR^2}{8}\omega + MR^2\Omega = 0$$

Step 6: Solve for the angular speed of the large disc.

Group the terms containing $\Omega$ and simplify.

$$\frac{3}{2}MR^2\Omega = -\frac{1}{8}MR^2\omega$$ $$\Omega = -\frac{\omega}{12}$$

Step 7: Determine the value of $n$.

Compare the magnitude of the angular speed to the given expression.

$$|\Omega| = \frac{\omega}{12} = \frac{\omega}{n}$$ $$n = 12$$

Answer: 12