Calculator Input
Assumed support model: pin at A and vertical roller at B. Positive angles are counterclockwise from the positive x-axis. Positive moments are counterclockwise.
Example Data Table
The sample below matches the default values loaded in the calculator.
| Item | Magnitude | Angle | x | y | Fx | Fy | Moment About A |
|---|---|---|---|---|---|---|---|
| Force 1 | 12 | -90° | 2 | 0 | 0.000 | -12.000 | -24.000 |
| Force 2 | 5 | 180° | 4 | 1 | -5.000 | 0.000 | 5.000 |
| Force 3 | 8 | 30° | 1 | 2 | 6.928 | 4.000 | -9.856 |
| External Moment | 10 N·m, counterclockwise | ||||||
| Support Reactions | A_x = -1.928 N, A_y = 4.857 N, B_y = 3.143 N | ||||||
| Equilibrium Verdict | Equilibrium satisfied within the stated tolerance. | ||||||
Formula Used
Each applied force is resolved into orthogonal components before the equilibrium equations are solved.
- Fx = F cos(θ)
- Fy = F sin(θ)
- MA = (x - xA)Fy - (y - yA)Fx
- ΣFx = 0
- ΣFy = 0
- ΣMA = 0
- Ax = -ΣFx
- By = -(ΣMA, applied + Mexternal) / (xB - xA)
- Ay = -ΣFy - By
This model assumes Support A provides horizontal and vertical reactions. Support B provides only a vertical reaction.
How to Use This Calculator
- Enter the coordinates of Support A and Support B.
- Type the external applied moment if one exists.
- Add the force magnitude, angle, and application point for each load.
- Choose a tolerance and optional unit labels.
- Click the calculate button to see reactions, totals, and checks.
- Use the CSV or PDF button to export the current result set.
About This Equilibrium of a Rigid Body Calculator
Rigid body equilibrium is a core topic in physics and engineering. A body stays in equilibrium when all forces balance and all turning effects balance. This calculator helps you test that condition quickly. It handles horizontal force balance, vertical force balance, and rotational balance about a support point.
Why Equilibrium Matters
Static equilibrium is used in beams, ladders, brackets, cranes, signs, and machine parts. It shows whether a structure can remain at rest without sliding or rotating. A correct equilibrium check also helps you estimate unknown support reactions. That makes design review faster and more reliable.
What This Tool Calculates
This page resolves each applied force into horizontal and vertical components. It then adds those components to find the total applied force. Next, it calculates the moment of each force about Support A. The tool combines all moments with any external applied moment. After that, it solves for the reaction forces at a pin support and a vertical roller support.
Practical Learning Benefits
Students can use this calculator to verify homework steps. Teachers can use it for quick classroom demonstrations. Engineers can use it to perform early design screening. Because every result is shown in a structured table, you can inspect each contribution before trusting the final answer.
Better Input Decisions
Use a clear sign convention before entering values. Positive angles are measured counterclockwise from the positive x-axis. Positive moments are counterclockwise. Keep units consistent for force and distance. If you use newtons and meters, moments will be in newton-meters.
When the Body Is Balanced
A rigid body is in equilibrium when the sum of horizontal forces equals zero, the sum of vertical forces equals zero, and the sum of moments equals zero. Small rounding differences can appear in decimal work. That is why this calculator includes a tolerance check to decide whether equilibrium is satisfied.
It also helps compare manual free-body diagrams with computed values. That reduces sign mistakes. Repeated trials with changed loads can reveal how sensitive stability is to force direction, support spacing, and load position.
This tool is useful for physics practice, statics revision, and support reaction estimation.
Frequently Asked Questions
1. What does equilibrium mean for a rigid body?
It means the body has no net horizontal force, no net vertical force, and no net turning effect. When all three conditions are satisfied, the body can remain at rest without translating or rotating.
2. Which support system does this calculator assume?
This version assumes a pin support at A and a vertical roller support at B. The pin produces horizontal and vertical reactions. The roller produces only a vertical reaction.
3. Can I enter angled forces?
Yes. Enter the force magnitude and angle in degrees. The tool converts that load into horizontal and vertical components automatically before calculating the reactions and equilibrium checks.
4. Why must Support B have a different x-coordinate from Support A?
The reaction at B is found from the moment equation about A. If both supports share the same x-coordinate, the lever arm becomes zero and this support model cannot solve for the vertical roller reaction.
5. What sign convention is used here?
Positive x is to the right. Positive y is upward. Positive angles rotate counterclockwise from the positive x-axis. Positive applied moments are also counterclockwise.
6. Does this calculator handle distributed loads directly?
Not directly. Convert each distributed load into an equivalent resultant force and apply it at the correct centroid location. Then enter that equivalent force into the calculator.
7. Why is the body sometimes shown as not balanced?
Small residual values can appear because of decimal rounding. If the checks exceed your chosen tolerance, the calculator marks the system as not balanced. Try increasing precision or reviewing load signs.
8. Which units should I use?
Use any consistent unit set. For example, use newtons and meters or pounds and feet. The unit labels are displayed in the result tables and export files for clarity.