Understanding the Ratio of Distance to Displacement for a Body Moving in a Semicircular Path

When a body moves along a semicircular path, the concepts of distance and displacement become crucial. Understanding these terms and how they relate to each other can provide deeper insights into the dynamics of motion. This article delves into the relationship between distance and displacement in a semicircular path and provides examples to illustrate the practical application of these principles.

Defining Distance and Displacement

Distance is defined as the total length of the path traveled by the body. In the case of a semicircular path, the distance is calculated as half the circumference of the full circle. The formula for distance in a semicircular path is given by:

[ text{Distance} frac{1}{2} times 2pi r pi r ]

Displacement, on the other hand, refers to the straight-line distance from the initial position to the final position. For a semicircular path, the initial and final positions are at the ends of the diameter. Therefore, the displacement is equal to the diameter of the circle:

[ text{Displacement} 2r ]

Calculating the Ratio of Distance to Displacement

To find the ratio of distance to displacement, simply divide the distance by the displacement:

[ text{Ratio} frac{text{Distance}}{text{Displacement}} frac{pi r}{2r} frac{pi}{2} ]

Thus, the ratio of distance to displacement when a body moves in a semicircular path is ( frac{pi}{2} ).

Example Calculation

Consider a specific instance where a body covers a distance of 4 meters along a semicircular path. To find the magnitude of displacement, use the following steps:

Distance 4 meters

Step 1: Calculate the radius of the semicircle using the distance covered:

[ text{Radius} frac{4}{pi} approx frac{4}{3.14} 1.274 , text{m} ]

Step 2: Calculate the diameter (displacement) of the semicircle:

[ text{Displacement} 2r 2 times 1.274 2.548 , text{m} ] (approximately)

Step 3: Calculate the ratio of distance to displacement:

[ text{Ratio} frac{4}{2.548} approx 1.571 text{ or } frac{pi}{2} approx 1.571 ]

This example demonstrates how the ratio of distance to displacement remains consistent with the general principle derived earlier.

Using Assumed Values for Pi

In practical scenarios, approximate values of π are often used for calculations. For instance, if π is taken as ( frac{22}{7} ) (a common approximation), the calculations would proceed as follows:

Step 1: Calculate the radius:

[ text{Radius} frac{4}{frac{22}{7}} frac{4 times 7}{22} frac{28}{22} frac{14}{11} , text{m} ]

Step 2: Calculate the displacement:

[ text{Displacement} 2 times frac{14}{11} frac{28}{11} , text{m} ] (or approximately 2.545 m)

Step 3: Calculate the ratio of distance to displacement:

[ frac{text{Distance}}{text{Displacement}} frac{4}{frac{28}{11}} frac{4 times 11}{28} frac{44}{28} frac{11}{7} ]

General Case with Unit Circle

For a unit circle where the radius ( r 1 ), the distance along the semicircle is given by:

[ text{Distance} pi r pi times 1 pi ]

The displacement is the diameter of the unit circle:

[ text{Displacement} 2r 2 times 1 2 ]

The ratio of distance to displacement is then:

[ text{Ratio} frac{pi}{2} ]

Curvature and Angle Relationship

Another way to understand the relationship is to consider the angle θ subtended at the center of the circle by the semicircular path. The ratio of distance to displacement in this context is given by:

[ text{Ratio} frac{theta}{sqrt{sin^2theta - (1 - costheta)^2}} ]

This expression simplifies to:

[ text{Ratio} frac{theta}{sqrt{2 - 2costheta}} ]

This relationship highlights the dependence of the ratio on the angle θ.

Conclusion

In summary, the ratio of distance to displacement when a body moves in a semicircular path is ( frac{pi}{2} ). This ratio remains consistent regardless of the specific values used for distance and displacement, making it a fundamental concept in understanding motion along curved paths. By applying this principle, one can analyze and predict the behavior of moving objects in semi-circular trajectories.