How to Find the Resultant Force of Three Forces?

To find the resultant force of three forces, you sum them together as vectors.

The resultant force, also known as the net force, is the single force vector that represents the combined effect of all individual forces acting on an object. According to the principles of vector addition, the net force is the vector sum of all the forces. That is, the net force is the resultant of all the forces; it is the result of adding all the forces together as vectors. For the situation of the three forces on the force board, the net force is the sum of force vectors A + B + C. This means if you have three forces, F₁, F₂, and F₃, the resultant force R is given by:

R = F₁ + F₂ + F₃ (as vectors)

Vector addition is different from simple arithmetic addition because forces have both magnitude and direction. There are two primary methods for finding the resultant of multiple forces:

Methods for Finding the Resultant Force

1. Graphical Method

This method involves drawing the forces to scale and then adding them graphically.

• Tail-to-Head Rule:

  • Draw the first force vector (F₁) starting from a point.
  • From the arrowhead (head) of the first vector, draw the tail of the second force vector (F₂).
  • From the arrowhead of the second vector, draw the tail of the third force vector (F₃).
  • The resultant vector (R) is drawn from the tail of the first vector to the arrowhead of the last vector.
  • The length of this resultant vector (using your chosen scale) gives its magnitude, and the angle it makes with a reference direction gives its direction.

• Polygon Method: The tail-to-head method for adding multiple vectors forms a polygon (or open polygon if the resultant is not zero). This is essentially the same process as the tail-to-head rule extended for more than two vectors.

While visually intuitive, the graphical method is less precise due to the accuracy limitations of drawing and measuring.

2. Analytical Method (Component Method)

This is the most accurate method and is commonly used in physics and engineering. It involves breaking down each force vector into its perpendicular components (usually along the x and y axes).

• Step 1: Resolve Each Force into Components:

  • For each force Fᵢ, find its components along the x-axis (Fᵢₓ) and the y-axis (Fᵢᵧ). If the force makes an angle θᵢ with the positive x-axis, the components are typically:
    • Fᵢₓ = Fᵢ cos(θᵢ)
    • Fᵢᵧ = Fᵢ sin(θᵢ)
  • Remember: Pay attention to the sign of the components based on the quadrant the force is in.

• Step 2: Sum the Components Along Each Axis:

  • Find the total resultant component along the x-axis (Rₓ) by adding the x-components of all forces:
    • Rₓ = F₁ₓ + F₂ₓ + F₃ₓ
  • Find the total resultant component along the y-axis (Rᵧ) by adding the y-components of all forces:
    • Rᵧ = F₁ᵧ + F₂ᵧ + F₃ᵧ

• Step 3: Find the Magnitude of the Resultant Force:

  • Use the Pythagorean theorem with the total x and y components:
    • Magnitude |R| = √(Rₓ² + Rᵧ²)

• Step 4: Find the Direction of the Resultant Force:

  • Use the arctangent function to find the angle (φ) the resultant makes with the positive x-axis:
    • φ = arctan(Rᵧ / Rₓ)
  • Important: You may need to adjust the angle based on the signs of Rₓ and Rᵧ to get the correct angle in the appropriate quadrant.

Example Insight:

Let's say you have three forces:

• F₁ = 10 N at 0° (along the positive x-axis)
• F₂ = 15 N at 90° (along the positive y-axis)
• F₃ = 20 N at 180° (along the negative x-axis)

Using the component method:

Force	Magnitude (N)	Angle (°)	x-component (Fᵢcosθ)	y-component (Fᵢsinθ)
F₁	10	0	10 cos(0°) = 10	10 sin(0°) = 0
F₂	15	90	15 cos(90°) = 0	15 sin(90°) = 15
F₃	20	180	20 cos(180°) = -20	20 sin(180°) = 0

Summing the components:

• Rₓ = 10 + 0 + (-20) = -10 N
• Rᵧ = 0 + 15 + 0 = 15 N

Finding the magnitude:

• |R| = √((-10)² + (15)²) = √(100 + 225) = √325 ≈ 18.03 N

Finding the direction:

• φ = arctan(15 / -10) = arctan(-1.5) ≈ -56.3°
• Since Rₓ is negative and Rᵧ is positive, the resultant is in the second quadrant. The angle with the positive x-axis is 180° - 56.3° = 123.7°.

So, the resultant force is approximately 18.03 N at an angle of 123.7° from the positive x-axis.

In summary, finding the resultant force of three forces involves adding them as vectors, which can be done accurately using the component method by resolving each force into perpendicular components, summing the components along each axis, and then calculating the magnitude and direction of the resultant from these sums.