Finding the Resultant Force and Y Component When a Force Acts at 60 Degrees

Understanding the mechanics of forces acting at angles is fundamental in physics and engineering. This article will guide you through the process of finding the resultant force and the y component of a force when the x component is known and the angle with the x axis is 60 degrees.

Introduction to Force Components

In this scenario, we have a force that is acting at an angle of 60 degrees with the x axis. The x component of this force is given as 50N. By using trigonometric relationships, we can determine the magnitude of the resultant force and the y component of the force.

Step-by-Step Solution

Step 1: Finding the Magnitude of the Force

The x component of the force is related to the magnitude of the force (F) by the equation:

Fx F ? cosθ

Given the angle θ 60 degrees and the x component Fx 50N, we can substitute the values to solve for F:

50 F ? cos60°

Sin(60°) 0.5, therefore:

50 F ? 0.5

Solving for F, we get:

F 50 / 0.5 100N

Step 2: Finding the y Component of the Force

The y component of the force can be found using the relationship:

Fy F ? sinθ

Substituting the known values and considering that sin(60°) √3/2, we get:

Fy 100 ? (√3/2) 50√3 ≈ 86.6N

Summary of Results

Resultant Force: 100N Y Component of the Force: approximately 86.6N

Theoretical Explanation with Right Triangles

The 30-60-90 triangle plays a crucial role in these calculations. In a 30-60-90 triangle:

The side opposite the 30° angle is half the length of the hypotenuse. The side opposite the 60° angle is √3/2 times the length of the hypotenuse.

Given the resultant force (hypotenuse) is 100N and the x component (adjacent side) is 50N, the remaining side (y component) can be calculated as:

y 86.6N (approx)

The cos 60° ≈ 0.866, and using the Pythagorean theorem, the magnitude of the resultant force can be calculated as:

R √(502 86.62) ≈ 100N

Conclusion

By following the steps and using trigonometric relationships, we can accurately determine the resultant force and the y component of the force when the x component is known and the angle of the force is 60 degrees. Understanding these principles is essential in a variety of fields, including physics, engineering, and computer graphics.