91Ó°ÊÓ

The Pythagorean theorem is a fundamental concept in mathematics, particularly useful in physics when dealing with vector quantities and right triangles. This theorem relates the squares of the lengths of the sides of a right triangle.In our exercise, we encounter a force vector \(\overrightarrow{\mathbf{F}}\) that has two components: a horizontal component \(F_x\) and a vertical component \(F_y\). These two components, along with the resultant force, \(\overrightarrow{\mathbf{F}}\), form a right triangle.To find the magnitude of \(\overrightarrow{\mathbf{F}}\), imagined as the hypotenuse of our triangle, we apply the Pythagorean theorem:

• \[ F = \sqrt{F_x^2 + F_y^2} \]
• Plugging in our values: \[ F = \sqrt{150^2 + 130^2} = \sqrt{39400} \]
• So, the magnitude is approximately \(198.49\,\text{N}\)

The beauty of the Pythagorean theorem lies in its simplicity and wide applicability in various problems involving right triangles.

Trigonometric functions are essential in solving problems that involve angles and ratios in right triangles. In this context, they help us find angles when dealing with vector components.To find the angle \(\theta\) in our exercise, we use the tangent function. Tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Thus, for our force components:

• \[ \tan(\theta) = \frac{F_y}{F_x} \]
• Inserting the given values: \[ \tan(\theta) = \frac{130}{150} = 0.8667 \]

To find \(\theta\), we calculate the inverse tangent (arctan) of this ratio:

• \[ \theta = \tan^{-1}(0.8667) \approx 40.6^\circ \]

Understanding these functions will not only help solve this problem but also assist in numerous applications involving vectors and angles.

Force components are critical to understand when analyzing forces in physics, especially when they act in more than one dimension. In this exercise, the force \(\overrightarrow{\mathbf{F}}\) is divided into two perpendicular parts:

• A horizontal component \(F_x = 150\,\text{N}\)
• A vertical component \(F_y = 130\,\text{N}\)

These components can be used to determine the overall force vector's characteristics, such as its magnitude and direction. The components help illustrate how forces can be separated into parts that act in specific directions, which simplifies the analysis of complex systems.Recognizing how to break down forces into components and understanding their significance allows students to tackle various physics problems with confidence and ease.