A coin is placed on a horizontal platform, which undergoes horizontal simple harmonic motion about a mean position O. The coin does not slip on the platform. The force of friction acting on the coin is F.
(i) F is always directed towards O
(ii) F is directed towards O when the coin is moving away from O, and away from O when the coin moves towards O
(iii) `F=0` when the coin and platform come to rest momentarily at the extreme position of the harmonic motion
(iv) F is maximum when the coin and platform come to rest momentarily at the extreme position of the harmonic motion

Explanation

To solve the problem, we need to analyze the motion of the coin on the platform that is undergoing simple harmonic motion (SHM). Let's break down the problem step by step:

### Step 1: Understanding the Motion
The platform is oscillating about a mean position O, which means it moves back and forth. The coin is placed on this platform and does not slip, indicating that it moves with the platform.

**Hint:** Consider how the motion of the platform affects the coin's motion.

### Step 2: Analyzing Forces on the Coin
As the platform moves, it accelerates towards the mean position O. The force of friction (F) is what keeps the coin from slipping off the platform. This frictional force must act in the direction of the acceleration of the platform.

**Hint:** Remember that friction always acts to oppose relative motion.

### Step 3: Direction of the Frictional Force
When the platform moves away from the mean position O, the coin tends to lag behind due to inertia. The frictional force must act towards O to accelerate the coin back towards the mean position.

Conversely, when the platform moves towards O, the coin is pushed along with it, and the frictional force acts in the opposite direction to prevent the coin from slipping off.

**Hint:** Think about the relationship between the direction of motion and the direction of friction.

### Step 4: Evaluating Each Option
- **(i)** F is always directed towards O: This is true because the frictional force always acts to bring the coin back to the mean position.
- **(ii)** F is directed towards O when the coin is moving away from O, and away from O when the coin moves towards O: This is false; F always acts towards O.
- **(iii)** F = 0 when the coin and platform come to rest momentarily at the extreme position of the harmonic motion: This is false; at the extreme position, the coin is momentarily at rest, but the frictional force is maximum to prevent slipping.
- **(iv)** F is maximum when the coin and platform come to rest momentarily at the extreme position of the harmonic motion: This is true because the maximum frictional force occurs when the platform is at the extremes of its motion.

### Conclusion
Based on the analysis, the correct answer is:
**(i) F is always directed towards O.**