Resultant Forces

1D Calculation (Collinear Forces)

• Same Direction: Add the magnitudes together. $F_{res} = F_1 + F_2$.
• Opposite Directions: Subtract the smaller magnitude from the larger. The direction of the resultant is the same as the larger force. $F_{res} = |F_1 - F_2|$.

2D Calculation (Non-Collinear Forces)

• Parallelogram Method: Draw two forces from a single point. Complete the parallelogram; the diagonal starting from the origin point represents the resultant force.
• Tip-to-Tail Method: Draw the first force vector, then draw the second starting from the tip of the first. The resultant is the vector from the start of the first to the tip of the last.
• Resolution of Forces: Break a diagonal force into two perpendicular components (usually horizontal $F \cos \theta$ and vertical $F \sin \theta$) to simplify complex systems.

• Resultant vs. Component: A resultant is the sum of parts, while a component is a piece of a single force broken down into specific directions (like x and y).

• Always Define Direction: In 1D problems, assign one direction as positive (e.g., Right = +) and the other as negative (Left = -). This prevents sign errors during subtraction.

• Check for Equilibrium: If a question states an object is moving at a 'steady speed' or is 'stationary', the resultant force must be zero. Use this to find missing force values.

• Scale Drawing Precision: When using scale diagrams, use a sharp pencil and a ruler. A $1 \text{ mm}$ error on a $1 \text{ cm} : 10 \text{ N}$ scale results in a $1 \text{ N}$ error in your final answer.

• Sanity Check: The resultant of two forces can never be larger than their sum or smaller than their difference. If $F_1 = 5 \text{ N}$ and $F_2 = 3 \text{ N}$, your resultant must be between $2 \text{ N}$ and $8 \text{ N}$.