A particle is moving with constant speed v on a circular path of 'r' radius when it has moved by angle `60^(@)`, Find
(i) Displacement of particle
(ii) Average velocity
(iii) Average acceleration

Answer

To solve the problem step by step, let's break it down into three parts as requested: displacement, average velocity, and average acceleration.

### Given:
- Radius of the circular path: \( r \)
- Angle moved by the particle: \( 60^\circ \)
- Constant speed of the particle: \( v \)

### (i) Displacement of the particle

1. **Understanding the Geometry**:
- The particle moves along a circular path and subtends an angle of \( 60^\circ \) at the center of the circle.
- The initial position of the particle is at point A, and the final position after moving \( 60^\circ \) is at point B.

2. **Drawing the Triangle**:
- The triangle formed by the center of the circle (O) and the two positions of the particle (A and B) is an isosceles triangle with OA = OB = r (the radius).

3. **Finding the Length of the Chord (Displacement)**:
- The angle at the center (O) is \( 60^\circ \).
- The displacement (AB) can be calculated using the formula for the chord length in a circle:
\[ AB = 2r \sin\left(\frac{\theta}{2}\right) \]
- Here, \( \theta = 60^\circ \):
\[ AB = 2r \sin\left(30^\circ\right) = 2r \cdot \frac{1}{2} = r \]
- **Thus, the displacement of the particle is \( r \)**.

### (ii) Average Velocity

1. **Understanding Average Velocity**:
- Average velocity is defined as the total displacement divided by the total time taken.
- We already found the displacement \( AB = r \).

2. **Finding the Time Taken**:
- The time taken to move through \( 60^\circ \) can be calculated using the formula:
\[ t = \frac{\text{Angle in radians}}{\text{Angular velocity}} = \frac{\theta}{\omega} \]
- The angular velocity \( \omega \) can be defined as:
\[ \omega = \frac{v}{r} \]
- Converting \( 60^\circ \) to radians:
\[ 60^\circ = \frac{\pi}{3} \text{ radians} \]
- Therefore, the time taken is:
\[ t = \frac{\frac{\pi}{3}}{\frac{v}{r}} = \frac{\pi r}{3v} \]

3. **Calculating Average Velocity**:
- Average velocity \( \vec{V}_{avg} \) is given by:
\[ \vec{V}_{avg} = \frac{\text{Displacement}}{\text{Time}} = \frac{r}{\frac{\pi r}{3v}} = \frac{3v}{\pi} \]
- **Thus, the average velocity is \( \frac{3v}{\pi} \)**.

### (iii) Average Acceleration

1. **Understanding Average Acceleration**:
- Average acceleration is defined as the change in velocity divided by the time taken.

2. **Finding Change in Velocity**:
- The initial velocity \( \vec{V}_1 \) is directed along the tangent at point A, and the final velocity \( \vec{V}_2 \) is directed along the tangent at point B.
- The angle between \( \vec{V}_1 \) and \( \vec{V}_2 \) is \( 60^\circ \).
- The change in velocity can be calculated using the formula:
\[ |\Delta \vec{V}| = |\vec{V}_2 - \vec{V}_1| = \sqrt{|\vec{V}_1|^2 + |\vec{V}_2|^2 - 2|\vec{V}_1||\vec{V}_2|\cos(60^\circ)} \]
- Since \( |\vec{V}_1| = |\vec{V}_2| = v \):
\[ |\Delta \vec{V}| = \sqrt{v^2 + v^2 - 2v^2 \cdot \frac{1}{2}} = \sqrt{v^2} = v \]

3. **Calculating Average Acceleration**:
- Average acceleration \( \vec{a}_{avg} \) is given by:
\[ \vec{a}_{avg} = \frac{|\Delta \vec{V}|}{t} = \frac{v}{\frac{\pi r}{3v}} = \frac{3v^2}{\pi r} \]
- **Thus, the average acceleration is \( \frac{3v^2}{\pi r} \)**.

### Summary of Answers:
- (i) Displacement = \( r \)
- (ii) Average Velocity = \( \frac{3v}{\pi} \)
- (iii) Average Acceleration = \( \frac{3v^2}{\pi r} \)