Calculating the Magnitude of an Unknown Force in a Geometric Problem

Understanding the resultant force in various scenarios can be challenging, especially when the direction of one force is reversed. In this article, we'll delve into a specific problem and provide a detailed solution involving mathematical principles and geometric methods.

Problem Statement

Given:
- The resultant of two forces is 7 N.
- One of the forces is 5 N.
- When the direction of the 5 N force is reversed, the resultant force becomes 19 N.

The question is: What is the magnitude of the other force?

Initial Setup and Analysis

We denote the two forces as F1 and F2. Given the conditions:

F1 F2 7 N F1 F2 7 N F1 - F2 19 N (when the 5 N force is reversed)

We need to find the magnitude of F2.

Approach Using Geometric Principles

Given the problem, it is beneficial to understand it geometrically. We can use the cosine rule to solve this problem, as the problem involves vector addition and subtraction, which can be visualized as triangles.

Step 1: Understanding the Triangles

We have two triangles created by the forces. In the first triangle:

Sides are 5 N, 7 N, and the angle between them is θ.

In the second triangle:

Sides are 5 N, 19 N, and the angle between them is 180° - θ.

Step 2: Applying the Cosine Rule

The cosine rule, which applies to any triangle, is given by:

c2 a2 b2 - 2abcos(θ)

For the first triangle (resultant 7 N), we get:

72 52 F22 - 2(5)F2cos(θ)

For the second triangle (resultant 19 N, with the direction reversed), we get:

192 52 F22 - 2(5)F2cos(180° - θ)

Note that cos(180° - θ) -cos(θ).

Step 3: Solving the Equations

Let's write both equations. First equation (7 N resultant):

49 25 F22 - 10F2cos(θ)

Second equation (19 N resultant):

361 25 F22 10F2cos(θ)

Let's simplify these equations:

49 25 F22 - 10F2cos(θ)
F22 - 10F2cos(θ) 24 361 25 F22 10F2cos(θ)
F22 10F2cos(θ) 336

Adding these two equations:

2F22 360
F22 180
F2 √180
F2 13.42 N (approximately)

Conclusion

The magnitude of the other force is approximately 13.42 N. This solution uses the geometric approach and the cosine rule to solve the problem, providing a clear and mathematical way to determine the unknown force.