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How to Find the Vector Sum of Forces

Published in Vector Addition of Forces 6 mins read

To find the vector sum of forces, also known as the net force, you combine all the individual force vectors acting on an object, taking into account both their magnitudes and directions. This resulting vector tells you the overall effect of all forces and is directly related to the object's acceleration, as highlighted in the provided reference.

Understanding Forces as Vectors

Forces are vector quantities, meaning they possess both a magnitude (how strong the force is) and a direction (which way it is pushing or pulling). When multiple forces act on an object, their combined effect is found by adding them as vectors, not just adding their magnitudes.

Steps to Find the Vector Sum

Finding the vector sum typically involves these steps:

Step 1: Identify All Forces Acting on the Object

The first crucial step is to determine every individual force acting on the object. As mentioned in the reference, "we can use free body diagrams to identify the force vectors." A free body diagram (FBD) is a visual tool where the object is represented as a point or a simple shape, and arrows are drawn from the object representing each force acting on it, indicating its direction and relative magnitude.

  • Examples of common forces include gravity, applied forces (pushes or pulls), normal force, friction, and tension.
  • Draw your FBD clearly, labeling each force vector.

Step 2: Choose a Method for Vector Summation

Once the forces are identified, you need to add these vectors together. There are two primary methods:

Method 1: Graphical Addition

This method involves drawing the force vectors to scale and using geometrical rules to find the resultant vector.

  • Tail-to-Head Method: Draw the first vector. Then, draw the second vector starting from the head (arrow end) of the first vector. Continue this for all vectors. The resultant vector is drawn from the tail of the first vector to the head of the last vector.
  • Parallelogram Method: Useful for adding two vectors. Place the tails of both vectors at the same point. Draw a parallelogram using these two vectors as adjacent sides. The diagonal from the common tail represents the resultant vector.
  • Pros: Good for visualization and understanding.
  • Cons: Can be less precise, especially with many vectors or complex angles, as it relies on accurate drawing and measurement.

Method 2: Analytical Addition (Component Method)

This is the most common and precise method, especially for multiple forces acting at various angles.

  1. Resolve Each Force into Components: Choose a coordinate system (usually x and y axes). Break down each force vector into its components along these axes. Use trigonometry (sine and cosine) to find the x and y components based on the force's magnitude and its angle relative to the axes.
    • If a force $\vec{F}$ has magnitude $F$ and makes an angle $\theta$ with the positive x-axis:
      • $F_x = F \cos(\theta)$
      • $F_y = F \sin(\theta)$
    • Pay attention to the signs of the components based on the quadrant the vector is in.
  2. Sum the Components Along Each Axis: Add all the x-components together to find the net x-component ($\Sigma F_x$), and add all the y-components together to find the net y-component ($\Sigma F_y$).
    • $\Sigma Fx = F{1x} + F{2x} + F{3x} + ...$
    • $\Sigma Fy = F{1y} + F{2y} + F{3y} + ...$
  3. Calculate the Magnitude of the Resultant Vector: The net force vector ($\vec{F}_{net}$) has components $\Sigma F_x$ and $\Sigma Fy$. Use the Pythagorean theorem to find the magnitude ($F{net}$) of the net force:
    • $F_{net} = \sqrt{(\Sigma F_x)^2 + (\Sigma F_y)^2}$
  4. Calculate the Direction of the Resultant Vector: Use the arctangent function to find the angle ($\phi$) of the net force relative to the x-axis:
    • $\phi = \arctan \left( \frac{\Sigma F_y}{\Sigma F_x} \right)$
    • Be careful to consider the signs of $\Sigma F_x$ and $\Sigma F_y$ to determine the correct quadrant for the angle.

The Significance of the Vector Sum

The vector sum of forces is crucial because it represents the net force ($\vec{F}{net}$) acting on the object. According to Newton's Second Law of Motion ($\vec{F}{net} = m\vec{a}$), the net force acting on an object is equal to the mass ($m$) of the object multiplied by its acceleration ($\vec{a}$).

Therefore, "use a vector sum to figure out the overall acceleration of the object," as the reference states. If the net force is zero, the object is in equilibrium (either at rest or moving at a constant velocity). If the net force is non-zero, the object will accelerate in the direction of the net force.

Example: Analytical Method

Let's find the vector sum of two forces acting on an object:

  • $\vec{F}_1$: 50 N at 30° above the positive x-axis.
  • $\vec{F}_2$: 70 N at 120° above the positive x-axis.

  1. Resolve into components:

    • $\vec{F}_1$:
      • $F_{1x} = 50 \cos(30°) = 50 \times 0.866 = 43.3 N$
      • $F_{1y} = 50 \sin(30°) = 50 \times 0.500 = 25.0 N$
    • $\vec{F}_2$:
      • $F_{2x} = 70 \cos(120°) = 70 \times (-0.500) = -35.0 N$
      • $F_{2y} = 70 \sin(120°) = 70 \times 0.866 = 60.6 N$

  2. Sum components:

    • $\Sigma Fx = F{1x} + F_{2x} = 43.3 N + (-35.0 N) = 8.3 N$
    • $\Sigma Fy = F{1y} + F_{2y} = 25.0 N + 60.6 N = 85.6 N$

  3. Calculate magnitude of net force:

    • $F_{net} = \sqrt{(8.3 N)^2 + (85.6 N)^2} = \sqrt{68.89 + 7327.36} = \sqrt{7396.25} \approx 85.99 N$

  4. Calculate direction of net force:

    • $\phi = \arctan \left( \frac{85.6 N}{8.3 N} \right) \approx \arctan(10.31) \approx 84.45°$
    • Since $\Sigma F_x$ is positive and $\Sigma F_y$ is positive, the angle is in the first quadrant, which is correct.

So, the vector sum of the forces (net force) is approximately 86.0 N at 84.5° above the positive x-axis.

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