Calculating the Displacement-Distance Ratio for a Particle Moving in a Circle

Understanding the relationship between displacement and distance is crucial in various applications in physics and engineering. In this article, we will explore a specific problem involving a particle moving in a circular path. We will break down the calculation into several steps and explain each process thoroughly to enhance comprehension.

Problem Overview

The question is: A particle is moving along a circle and completes one revolution in 40 seconds. What is the ratio of displacement to distance traveled by the particle in 2 minutes and 20 seconds?

Step 1: Determine Angular Speed

The first step in solving this problem is to determine the angular speed of the particle. Since the particle completes one full revolution (2π radians) in 40 seconds, the angular speed (omega) can be calculated as follows:

(omega frac{2pi text{ radians}}{40 text{ seconds}} frac{pi}{20} text{ radians/second})

Step 2: Convert Total Time into Seconds

We need to convert the total time given (2 minutes and 20 seconds) into seconds. This is done as follows:

(2 text{ minutes} 2 times 60 120 text{ seconds}) (20 text{ seconds} 20 text{ seconds}) (text{Total time} 120 20 140 text{ seconds})

Step 3: Calculate the Number of Revolutions

Next, we find out how many full revolutions the particle completes in 140 seconds:

(text{Number of revolutions} frac{text{Total time}}{text{Time per revolution}} frac{140 text{ seconds}}{40 text{ seconds/revolution}} 3.5 text{ revolutions})

Step 4: Calculate the Distance Traveled

The distance traveled by the particle after 3.5 revolutions can be calculated as follows:

(text{Distance} text{Number of revolutions} times text{Circumference of the circle} 3.5 times 2pi r 7pi r)

Step 5: Calculate the Displacement

The displacement is the straight-line distance from the starting point to the ending point. After 3.5 revolutions, the particle completes 3 full revolutions and then moves an additional 0.5 revolution. Thus, the particle is at the opposite side of the circle from its starting point. The displacement will be the diameter of the circle:

(text{Displacement} 2r)

Here, the angle corresponding to 3.5 revolutions is:

(3.5 text{ revolutions} 3.5 times 2pi 7pi text{ radians})

After 3 full revolutions (6π radians) the particle is back at the starting point, and then it moves an additional π radians, effectively bringing it to the opposite side of the circle. This results in a straight-line distance equal to the diameter of the circle, which is 2r.

Step 6: Calculate the Ratio of Displacement to Distance

The final step is to find the ratio of displacement to distance:

(text{Ratio} frac{text{Displacement}}{text{Distance}} frac{2r}{7pi r} frac{2}{7pi})

Final Answer

Thus, the ratio of displacement to distance after 2 minutes and 20 seconds is:

(frac{2}{7pi})

Conclusion

Understanding the calculation of this ratio helps in grasping the fundamental concepts of displacement and distance. This problem serves as a clear example of how to determine and calculate these quantities in a circular motion scenario.

Key Points:

Angular speed is crucial in calculating the speed of a particle moving in a circular path. Total time needs to be converted into seconds to ensure consistency in calculations. The number of revolutions determines the distance traveled by the particle. Displacement is the straight-line distance from the starting point to the ending point. The ratio of displacement to distance can be useful in various practical applications.

By following these steps, we can accurately calculate the ratio of displacement to distance, enhancing our understanding of circular motion and its applications.