91Ó°ÊÓ

In physics, when multiple forces act on an object, each force can be broken down into parts, known as components. This technique simplifies analysis by translating forces from angled vectors into easier-to-calculate horizontal and vertical vectors.
Imagine a force acting at an angle. Instead of dealing with a diagonal pull or push, you break this force into two parts: one acting purely in the horizontal direction (x-component) and another purely in the vertical direction (y-component). This breakdown allows for straightforward calculations of effects on the object.
For example, if Force \(\overrightarrow{\mathbf{F}}_1\) has a magnitude of \(21.0 \,\text{N}\) and is angled at \(30^{\circ}\), we use trigonometric functions to find:

• x-component: \(\overrightarrow{\mathbf{F}}_{1x} = -21.0 \cos(30^{\circ})\)
• y-component: \(\overrightarrow{\mathbf{F}}_{1y} = 21.0 \sin(30^{\circ})\)

Breaking forces into components is essential for solving problems efficiently, especially when dealing with multiple forces like in our exercise.

The concept of equilibrium refers to the state where all the forces acting on an object balance each other out, resulting in no net force. This means the object remains at rest or moves at constant velocity without accelerating.
To achieve equilibrium, the sum of all forces acting along each axis (x and y) must be zero.
In our exercise, we have two known forces, and we need a third force \(\overrightarrow{\mathbf{F}}_3\) to ensure equilibrium. This means:

• The total force in the x-direction should be zero: \(\overrightarrow{\mathbf{F}}_{1x} + \overrightarrow{\mathbf{F}}_{2x} + \overrightarrow{\mathbf{F}}_{3x} = 0\)
• The total force in the y-direction should be zero: \(\overrightarrow{\mathbf{F}}_{1y} + \overrightarrow{\mathbf{F}}_{2y} + \overrightarrow{\mathbf{F}}_{3y} = 0\)

Equilibrium is key to ensuring stability in systems, whether in structures or physics problems.

Trigonometric functions such as sine, cosine, and tangent are invaluable tools in physics, particularly for analyzing forces and their components.
These functions help translate the magnitude and direction of a force into its perpendicular components. For example, the angle and magnitude of a force can establish how much of the force acts in a specific direction.
Calculating components like in the exercise involves using:

• Cosine for the adjacent side, which gives the x-component: \(\text{adjacent} = \text{hypotenuse} \times \cos(\theta)\)
• Sine for the opposite side, which contributes the y-component: \(\text{opposite} = \text{hypotenuse} \times \sin(\theta)\)

When angles are involved, especially in vectors on inclined planes or forces at an angle, trigonometric functions become the bridge for precise calculations.

The Pythagorean theorem is not just for mathematics; it plays a crucial role in physics, particularly concerning vectors and forces.
It states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
For forces, this theorem helps find the resultant vector's magnitude from its components. If you know the perpendicular x and y components, you can use:\[|\overrightarrow{\mathbf{F}}| = \sqrt{(\overrightarrow{\mathbf{F}}_x)^2 + (\overrightarrow{\mathbf{F}}_y)^2}\]In our exercise, we use it to determine the magnitude of \(\overrightarrow{\mathbf{F}}_3\) knowing both its x (\overrightarrow{\mathbf{F}}_{3x})(3.2 N) and y (\overrightarrow{\mathbf{F}}_{3y})(-10.5 N) components.
Using the Pythagorean theorem ensures accurate and reliable results when calculating overall forces from perpendicular parts.