Understanding Forces and Resultants: A Mathematical Analysis

When dealing with forces, especially when they are perpendicular to each other, the analysis becomes both interesting and applicable to real-world scenarios. This article will explore the concept of forces whose resultant is 100 N (Newtons) and delve into the mathematics behind determining their magnitudes. This is crucial for various fields such as physics, engineering, and even everyday problem-solving.

Introduction to Perpendicular and Resultant Forces

Two forces are said to be perpendicular when the angle between them is 90 degrees. In such cases, there is no component of one force in the direction of the other. When these forces combine, they form a resultant force, which is a single force that has the same effect as the combination of the individual forces. Mathematically, if we have two perpendicular forces, we can use vector addition to determine the resultant force and the individual magnitudes of the forces.

Calculating Magnitudes of Perpendicular Forces

Given that two forces are perpendicular, and the resultant of these forces is 100 N, we can use vector addition and trigonometric principles to find the magnitudes of these forces. Let's delve into the detailed steps.

Step 1: Define the Forces and Resultant

Let R be the resultant force, and F1 and F2 be the two perpendicular forces. Since the forces are perpendicular, the angle between them is 90 degrees. The resultant R makes an angle of 60 degrees with one of the forces, say F1.

Step 2: Use Trigonometry and Vector Addition

From the problem, we know that the resultant R is 100 N. Given that F1 makes an angle of 60 degrees with R, we can express the relationship between F1 and R using cosine:

F1 R cos 60°

Given that cos 60° 0.5, we have:

F1 100 N times; 0.5 50 N

Step 3: Use Pythagorean Theorem to Find F2

Using the Pythagorean theorem, we can find the magnitude of the other force F2

R2 F12 F22

Substituting the known values:

1002 502 F22

10000 2500 F22

Solving for F2:

F22 10000 - 2500 7500

F2 √7500 ≈ 86.60 N

Verification of the Results

Let's verify the results by checking if the components of the forces satisfy the given conditions. If F1 makes a 60° angle with the resultant and is perpendicular to F2, then the components of F1 and F2 in the direction of R should be consistent with the given resultant.

For F1, the component along the direction of R is:

F1 cos 60° 50 N

For F2, the component along the direction of R is:

F2 cos 30° 86.60 N times; 0.8660 ≈ 75.00 N

Adding these components:

50 N 75.00 N 100 N, which matches the resultant.

Conclusion

By using vector addition and trigonometric principles, we can accurately determine the magnitudes of forces when they are perpendicular and combine to form a resultant. The example provided demonstrates the application of these principles in a straightforward manner, offering insights into how to handle such problems in various scientific and engineering contexts.