Physics · Unit 2: Dynamics ·  Project 2-1-4

Deep Dive: Adding Force Vectors

🔬 Deep Dive
A new skill for Project 2-1-4. Take your time — this is real math, not just review.
AConcept

The Force Table Setup

A force table has a ring at its center, with strings running out to pulleys mounted around the edge. Each string goes over a pulley and down to a hanging weight — the weight's value IS the string's force in newtons, and the pulley's position sets that force's angle.

Angles are measured the standard way: counterclockwise from the positive x-axis (0°), just like a unit circle. A force at 90° points straight "up" on the table; a force at 180° points straight "left."

BMath

Breaking a Force Into Components

Any force at an angle can be split into two forces that add up to it: one pointing purely along the x-axis, one pointing purely along the y-axis.

Fx = F cosθ   |   Fy = F sinθ
⚠️Your calculator's cos and sin functions will happily hand back negative numbers for angles past 90° — that's correct, not a mistake. A negative Fx means that force's horizontal push is actually pointing in the negative x-direction. Don't second-guess the sign.
CMath

Adding Multiple Forces

To add several forces together, add their components separately — every x-component together, every y-component together.

ΣFx = Fx₁ + Fx₂ + ...   |   ΣFy = Fy₁ + Fy₂ + ...

The two totals, ΣFx and ΣFy, are the components of the resultant — the single force that has the same overall effect as all the individual forces combined.

DMath

Finding Magnitude and Angle

Once you have ΣFx and ΣFy, the Pythagorean theorem gives you the resultant's size, and inverse tangent gives you its direction.

|R| = √(ΣFx² + ΣFy²)   |   θ = tan⁻¹(ΣFy / ΣFx)
⚠️Your calculator's inverse tangent only ever returns an angle between −90° and 90° — it doesn't know which quadrant your resultant actually points into. Check the signs of ΣFx and ΣFy yourself and adjust the angle by 180° if needed so it lands in the correct quadrant.
EConcept

The Equilibrant: Finding the Third Force

The equilibrantis the single force that would perfectly cancel the resultant — same magnitude, opposite direction (add 180° to the resultant's angle).

equilibrant magnitude = |R|   |   equilibrant angle = θ_R + 180°

Add the equilibrant to the original two forces, and every component sums to exactly zero — which, by Newton's First Law, means the ring stays perfectly still.

Enter magnitude and angle (measured counterclockwise from the positive x-axis, like a standard force table) for each known force. This computes the resultant AND the equilibrant — the force that would balance them.

magnitude (N)angle (°)
F1
F2
F1F2REquilibrant
ΣFx  /  ΣFy
19.6 N  /  46.0 N
resultant (mag / angle)
50.0 N / 66.9°
Equilibrant: 50.0 N at 246.9°(same size as the resultant, pointing exactly the opposite way)

Once your teacher assigns your two real forces, come back here and plug in your actual numbers as your prediction — then copy the result to your project worksheet.

ExampleWorked Example — Two Known Forces

Force 1 is 40 N at 30°. Force 2 is 30 N at 120°. Find the equilibrant that balances both.

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