To solve the problem of a particle moving in a circle with a constant speed, we can break it down into two parts: finding the time period of the particle and calculating the average speed, average velocity, and average acceleration over a specified time interval. Let's tackle each part step by step.
Finding the Time Period
The time period (T) of a particle moving in a circular path is the time it takes to complete one full revolution. We can use the formula:
- Speed (v) = Distance (d) / Time (T)
For circular motion, the distance traveled in one complete revolution is the circumference of the circle, which can be calculated using the formula:
Given that the radius (r) is 4 cm, we can calculate the circumference:
Now, we can rearrange the speed formula to find the time period:
- T = Distance / Speed = C / v
Substituting the values:
- T = (8π cm) / (1 cm/sec) = 8π seconds
Calculating Average Speed, Average Velocity, and Average Acceleration
Next, we need to find the average speed, average velocity, and average acceleration over the time interval from t = 0 to t = T/4.
Average Speed
The average speed is defined as the total distance traveled divided by the total time taken. In this case, from t = 0 to t = T/4, the particle travels a quarter of the circumference:
- Distance traveled = (1/4) * C = (1/4) * (8π cm) = 2π cm
- Total time = T/4 = (8π seconds) / 4 = 2π seconds
Now we can calculate the average speed:
- Average Speed = Total Distance / Total Time = (2π cm) / (2π seconds) = 1 cm/sec
Average Velocity
Average velocity is defined as the total displacement divided by the total time. The displacement from the starting point to the point at T/4 is a straight line from the center of the circle to the point on the circumference, which forms a right triangle:
- Displacement = 4 cm (the radius, since it moves to the right at T/4)
- Total time = 2π seconds
Calculating average velocity:
- Average Velocity = Displacement / Total Time = (4 cm) / (2π seconds) = 2/(π) cm/sec
Average Acceleration
Average acceleration is defined as the change in velocity divided by the time interval. Since the particle is moving with constant speed, we need to consider the change in direction:
- Initial velocity (v₀) = 1 cm/sec (at t = 0)
- Final velocity (v₁) = 1 cm/sec (at t = T/4, but in a different direction)
The change in velocity can be calculated using vector components. The change in direction is 90 degrees, and we can use the formula:
- Change in velocity = v₁ - v₀ = 1 cm/sec (at 90 degrees)
Using the formula for average acceleration:
- Average Acceleration = Change in Velocity / Total Time = (1 cm/sec) / (2π seconds) = 1/(2π) cm/sec²
Summary of Magnitudes
To summarize the magnitudes we found:
- Time Period (T) = 8π seconds
- Average Speed = 1 cm/sec
- Average Velocity = 2/(π) cm/sec
- Average Acceleration = 1/(2π) cm/sec²
This breakdown should help you understand the concepts of circular motion and how to calculate these quantities effectively. If you have any further questions or need clarification on any point, feel free to ask!
