Calculate the moment of inertia and initial angular momentum of a turntable.

Given:
Mass $m = 2 \text{ kg}$
Diameter $D = 34 \text{ cm}$
Initial angular speed $\omega_0 = 3.5 \text{ rad/s}$
Final angular speed $\omega_f = 0 \text{ rad/s}$ (comes to rest)
Number of revolutions $N = 3$

First, convert the diameter to radius in meters and calculate the total angular displacement:
Radius $R = \frac{D}{2} = \frac{34 \text{ cm}}{2} = 17 \text{ cm} = 0.17 \text{ m}$
Angular displacement $\theta = N \times 2\pi \text{ rad} = 3 \times 2\pi = 6\pi \text{ rad}$

(iii) Moment of inertia of the turn table about its axis.

Step 1: Use the formula for the moment of inertia of a solid disk.
For a solid disk rotating about its central axis, the moment of inertia $I$ is given by:

$$I = \frac{1}{2} m R^2$$

Step 2: Substitute the given mass and calculated radius.

$$I = \frac{1}{2} (2 \text{ kg}) (0.17 \text{ m})^2$$

Step 3: Calculate the moment of inertia.

$$I = (1 \text{ kg}) (0.0289 \text{ m}^2)$$ $$I = 0.0289 \text{ kg} \cdot \text{m}^2$$

The moment of inertia is $\boxed{\text{0.0289 kg}\cdot\text{m}^2}$.

(i) Angular momentum

Step 1: Use the formula for angular momentum.
Angular momentum $L$ is given by $L = I \omega$. Since the question refers to an initial speed, we calculate the initial angular momentum $L_0$.

$$L_0 = I \omega_0$$

Step 2: Substitute the calculated moment of inertia and initial angular speed.

$$L_0 = (0.0289 \text{ kg} \cdot \text{m}^2) (3.5 \text{ rad/s})$$

Step 3: Calculate the initial angular momentum.

$$L_0 = 0.10115 \text{ kg} \cdot \text{m}^2/\text{s}$$

The initial angular momentum is $\boxed{\text{0.101 kg}\cdot\text{m}^2/\text{s}}$.

(ii) The time taken for it to come to rest after it is switched off.

Step 1: Calculate the angular acceleration ($\alpha$) using rotational kinematics.
We use the equation:

$$$$