To solve the problem step by step, we will break it down into parts (a) and (b).
### Part (a): Finding the Time Period of the Particle
1. **Identify the Formula for Time Period**:
The time period \( T \) of a particle moving in a circle is given by the formula:
\[
T = \frac{\text{Total Distance}}{\text{Speed}}
\]
The total distance for one complete revolution (circumference of the circle) is \( 2\pi r \).
2. **Substituting the Values**:
Given:
- Radius \( r = 4 \, \text{cm} \)
- Speed \( v = 1 \, \text{cm/s} \)
Now, calculate the circumference:
\[
\text{Circumference} = 2\pi r = 2 \times 3.14 \times 4 \, \text{cm} = 25.12 \, \text{cm}
\]
3. **Calculating Time Period**:
Substitute the values into the time period formula:
\[
T = \frac{25.12 \, \text{cm}}{1 \, \text{cm/s}} = 25.12 \, \text{s}
\]
Thus, the time period \( T \) is approximately \( 25.12 \, \text{s} \).
### Part (b): Finding Average Speed, Average Velocity, and Average Acceleration from \( t = 0 \) to \( t = \frac{T}{4} \)
1. **Average Speed**:
Since the particle is moving with constant speed, the average speed is equal to the instantaneous speed:
\[
\text{Average Speed} = v = 1 \, \text{cm/s}
\]
2. **Average Velocity**:
- **Displacement Calculation**:
At \( t = \frac{T}{4} \), the particle moves \( 90^\circ \) around the circle. The displacement can be calculated using the Pythagorean theorem. The displacement from point A (starting point) to point B (point after \( 90^\circ \)) is:
\[
\text{Displacement} = r\sqrt{2} = 4\sqrt{2} \, \text{cm}
\]
- **Time Interval**:
The time interval from \( t = 0 \) to \( t = \frac{T}{4} \) is:
\[
\Delta t = \frac{T}{4} = \frac{25.12 \, \text{s}}{4} = 6.28 \, \text{s}
\]
- **Calculating Average Velocity**:
\[
\text{Average Velocity} = \frac{\text{Displacement}}{\Delta t} = \frac{4\sqrt{2} \, \text{cm}}{6.28 \, \text{s}} \approx 0.9 \, \text{cm/s}
\]
3. **Average Acceleration**:
- **Change in Velocity**:
The initial velocity \( v_i = 1 \, \text{cm/s} \) and the final velocity \( v_f \) can be calculated using the average velocity. Since average velocity is less than the initial speed, we can assume that the final velocity is approximately \( 0.9 \, \text{cm/s} \).
- **Calculating Average Acceleration**:
\[
\text{Average Acceleration} = \frac{\Delta v}{\Delta t} = \frac{v_f - v_i}{\Delta t} = \frac{0.9 \, \text{cm/s} - 1 \, \text{cm/s}}{6.28 \, \text{s}} \approx -0.016 \, \text{cm/s}^2
\]
### Summary of Results:
- (a) Time Period \( T \approx 25.12 \, \text{s} \)
- (b) Average Speed \( = 1 \, \text{cm/s} \)
- Average Velocity \( \approx 0.9 \, \text{cm/s} \)
- Average Acceleration \( \approx -0.016 \, \text{cm/s}^2 \)
