Calculating Displacement In Circular Motion A Physics Guide

In the realm of physics, understanding motion is paramount. This exploration delves into a fascinating scenario involving an athlete running along the circumference of a circular field, a classic example used to illustrate the concepts of displacement and distance in physics. The problem at hand challenges us to calculate the magnitude of the athlete's displacement after completing three-quarters of a revolution around a circular field with a diameter of 20 meters. Before we dive into the solution, let's first clarify the key concepts involved: displacement, which is the shortest distance between the initial and final positions, and distance, which is the total length of the path traveled. The athlete's journey around the circular field provides an excellent context to differentiate these two fundamental concepts.

Keywords: Displacement, distance, circular motion, revolution, diameter, physics problem, athlete's journey, initial position, final position, magnitude.

To accurately calculate the athlete's displacement, it's crucial to grasp the distinct meanings of displacement and distance. Distance, a scalar quantity, measures the total path length covered during motion. In contrast, displacement, a vector quantity, signifies the shortest distance between the starting and ending points, taking direction into account. Imagine the athlete running three-quarters of the circular track; the distance covered would be three-quarters of the circle's circumference. However, displacement focuses solely on the straight-line distance from the athlete's starting point to their final location. This difference is vital in many physics problems, especially those involving circular motion, where an object can travel a considerable distance yet have a small displacement, or even zero displacement if it returns to its starting point. Understanding the interplay between distance and displacement allows us to analyze motion more comprehensively, predicting not only how far an object has moved but also its change in position relative to its origin. This distinction is fundamental in fields like navigation, sports science, and engineering, where precise calculations of movement and positional changes are critical. To further illustrate, consider an athlete completing a full lap around the circular field. The distance covered would be the entire circumference of the circle. However, the displacement would be zero because the athlete returns to the exact starting point. This example underscores the importance of considering the path taken (distance) versus the net change in position (displacement).

Keywords: Scalar quantity, vector quantity, total path length, shortest distance, starting point, ending point, direction, circular track, circumference, straight-line distance, net change in position, navigation, sports science, engineering.

The circular field in our problem has a diameter of 20 meters, a crucial piece of information that directly impacts our calculations. The diameter is the straight-line distance that passes through the center of the circle, connecting two points on the circumference. To find the radius, which is essential for calculating both circumference and displacement, we simply divide the diameter by two. Thus, the radius of the field is 10 meters. The athlete's journey covers three-quarters of the circular path. This detail is critical because it determines the final position relative to the starting point. After completing half a revolution (1/2), the athlete would be directly opposite the starting point. After completing a full revolution (1), the athlete would return to the starting point. However, at three-quarters (3/4) of the revolution, the athlete's final position forms a specific geometric relationship with the starting point and the center of the circle. Visualizing this trajectory is key to solving the problem. Imagine drawing a circle and marking the starting point. Then, trace three-quarters of the circle's circumference. The endpoint will be neither directly opposite the start nor back at the start, but at a point that creates a right angle with the starting point when connected through the circle's center. This right-angle relationship simplifies the displacement calculation, allowing us to use the Pythagorean theorem.

Keywords: Diameter, radius, circumference, starting point, final position, geometric relationship, circular path, revolution, right angle, Pythagorean theorem, visualize trajectory, center of the circle.

Now, let's calculate the magnitude of the athlete's displacement. As established, the athlete's path forms three-quarters of a circle, and the final position creates a right angle with the starting position when connected through the center. This scenario allows us to visualize the displacement as the hypotenuse of a right-angled triangle. The other two sides of this triangle are the radii of the circle, each measuring 10 meters. The Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides, is our key tool here. Mathematically, this is expressed as: a² + b² = c², where 'c' represents the hypotenuse (displacement in our case), and 'a' and 'b' are the other two sides (the radii). Plugging in the values, we get: 10² + 10² = c², which simplifies to 100 + 100 = c², or 200 = c². To find 'c', we take the square root of both sides: c = √200. This can be further simplified as c = √(100 * 2) = 10√2 meters. Therefore, the magnitude of the athlete's displacement after completing three-quarters of the revolution is 10√2 meters. This result highlights how the displacement, the straight-line distance, is less than the actual distance traveled along the circular path, emphasizing the critical difference between these two concepts in physics.

Keywords: Magnitude of displacement, right-angled triangle, hypotenuse, radii, Pythagorean theorem, square root, calculation steps, 10√2 meters, straight-line distance, actual distance, physics concepts.

In conclusion, the problem of the athlete running around the circular field vividly illustrates the concept of displacement and its distinction from distance. By applying geometric principles and the Pythagorean theorem, we determined that the magnitude of the athlete's displacement after completing three-quarters of the revolution is 10√2 meters. This exercise not only reinforces our understanding of displacement but also highlights the importance of visualizing motion and understanding the relationships between different physical quantities. Displacement, as a fundamental concept in physics, plays a crucial role in various applications, from navigation and sports analysis to advanced physics problems. Recognizing the difference between displacement and distance allows for more accurate analysis and prediction of motion, making it an essential skill for students and professionals alike. The athlete's circular journey serves as a powerful reminder that the shortest path between two points is not always the path traveled, and that understanding this difference is key to unlocking deeper insights into the world of physics. Further exploration of related topics such as velocity, acceleration, and vectors can build upon this foundation and enhance one's understanding of motion in even more complex scenarios.

Keywords: Significance of displacement, geometric principles, Pythagorean theorem, visualizing motion, physical quantities, navigation, sports analysis, velocity, acceleration, vectors, understanding motion, circular journey, fundamental concept.