A particle of mass `m` moves in a circle of radius `R` with a uniform speed `v`.
(i) the angular speed of particle is `v//R`.
(ii) the time period of revolution is `2piR//v`.
(iii) the acceleration of particle is `v^(2)//R`.
(iv) the work done by the centripetal force in half revolution is `(mv^(2)//R)xxpiR`.

Options

A. `(i),(ii)`
B. `(ii),(iii)`
C. `(ii),(ii),(iii)`
D. All options are correct

Explanation

To solve the problem step by step, we will analyze each statement regarding the motion of a particle of mass `m` moving in a circle of radius `R` with uniform speed `v`.

### Step 1: Angular Speed
The angular speed (ω) of a particle moving in a circle is given by the formula:

\[ \omega = \frac{v}{R} \]

Where:
- \( v \) is the linear speed of the particle.
- \( R \) is the radius of the circle.

Thus, the first statement is correct.

### Step 2: Time Period of Revolution
The time period (T) of one complete revolution is the time taken to travel the circumference of the circle. The circumference (C) of a circle is given by:

\[ C = 2\pi R \]

Since the speed is uniform, the time period can be calculated as:

\[ T = \frac{C}{v} = \frac{2\pi R}{v} \]

Thus, the second statement is correct.

### Step 3: Acceleration of the Particle
In uniform circular motion, the particle experiences centripetal acceleration (a_c), which is directed towards the center of the circle. The formula for centripetal acceleration is:

\[ a_c = \frac{v^2}{R} \]

Since the speed is constant, there is no tangential acceleration. Therefore, the third statement is correct.

### Step 4: Work Done by the Centripetal Force
The work done (W) by a force is given by the formula:

\[ W = F \cdot d \cdot \cos(\theta) \]

Where:
- \( F \) is the force,
- \( d \) is the displacement,
- \( \theta \) is the angle between the force and displacement.

In the case of centripetal force, it acts perpendicular to the direction of motion of the particle. Since the angle \( \theta = 90^\circ \), we have:

\[ \cos(90^\circ) = 0 \]

Thus, the work done by the centripetal force is:

\[ W = F \cdot d \cdot 0 = 0 \]

Therefore, the fourth statement is incorrect.

### Summary of Statements
1. Angular speed is \( \frac{v}{R} \) - **Correct**
2. Time period of revolution is \( \frac{2\pi R}{v} \) - **Correct**
3. Acceleration of the particle is \( \frac{v^2}{R} \) - **Correct**
4. Work done by the centripetal force in half revolution is \( \frac{mv^2}{R} \cdot \pi R \) - **Incorrect**