Understanding Vector Decomposition and Resultant Calculation

In physics, vector decomposition is a technique used to find the components of a vector that act at various angles. This is crucial in diverse fields such as engineering, navigation, and essentially anywhere forces or displacements interact. This article aims to elucidate a specific problem involving the determination of the second force and resultant angle, following a detailed analysis of the given forces and their interactions.

The Problem Scenario

We are faced with a scenario involving two forces: the first force is 195 N, and the second force is unknown. These forces act at an angle of 75 degrees. The resultant force from these two forces is 225 N. The task is to calculate the magnitude of the second force and the angle it makes with the resultant force.

Mathematical Formulation and Solution

To solve this problem, we utilize the law of cosines and basic trigonometry principles. Here's a step-by-step breakdown:

Step 1: Application of the Law of Cosines to Determine the Second Force

Given: Resultant force (R) 225 N, first force (F1) 195 N, and the angle between them (θ) 75° Let the second force be F2 Using the law of cosines: R2 F12 F22 - 2 * F1 * F2 * cos(180° - θ) Note: cos(180° - 75°) -cos(75°) Substitute the known values: 2252 1952 F22 - 2 * 195 * F2 * (-cos(75°)) This simplifies to: 12600 F22 391.8787451 * F2 Reorganize into a quadratic equation: F22 391.8787451 * F2 - 12600 0 Solve the quadratic equation using the quadratic formula: F2 -b ± √(b2-4ac) / 2a Here, a 1, b 391.8787451, and c -12600 Calculate the solutions: F2 -195.9394276 ± √(195.93942762 - 4 * 1 * -12600) / 2 The positive solution is F2 72.60421525 N

Hence, the second force has a magnitude of approximately 72.6 N.

Step 2: Angle Between Forces

To find the angle between the 195 N force and the resultant force, we use another application of the law of cosines: B cos-1(F12 R2 - F22) / (2 * F1 * R) Substitute the known values: B cos-1(1952 2252 - 72.604215252) / (2 * 195 * 225) Simplify: B cos-1(83378.62793 / 87750) B ≈ 18.1611171° or ~18.16°

Step 3: Resultant Angle Calculation

The angle between the second force and the resultant force is calculated as:

Resultant angle (φ) 75° - 18.16° 56.84°

Conclusion

Through the application of vector analysis principles, specifically the law of cosines, we have successfully determined the magnitude of the second force (72.6 N) and its resultant angle with respect to the resultant force (56.84°). This process showcases the importance of vector analysis in solving real-world physics problems involving forces and angles.

For further exploration, you may refer to the following resources for a deeper understanding of vector decomposition and resultant force calculation:

Physics Classroom – Vector Addition Khan Academy – Vector Components