1. Title and aim
General static equilibrium of non-concurrent coplanar parallel forces. Investigate the translational and rotational equilibrium conditions for non-concurrent parallel coplanar forces acting on a rigid beam supported at two points, verify that the sum of support reactions equals the total downward load (Σ F_y = 0) and the algebraic sum of moments vanishes about any axis (Σ τ = 0), and determine the unknown concentrated load weight and beam mass using ordinary least-squares regression.
2. Abstract
Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of concentrated load weight of 5.0000 N, differing by 0.000% from the model value. The reported scatter comes from scale rounding and any stated model approximation, not measured laboratory noise.
3. Apparatus and fixed settings
- Uniform graduated rigid beam: A seasoned 1.000 m hardwood beam of rectangular cross-section, engraved with millimeter scale graduations along its spine, exhibiting a verified center of gravity at x_cg = 0.500 m and a uniform self-weight W_bar = 2.000 N.
- Twin overhead tubular dynamometers: Two precision tubular spring dynamometers (0 to 10 N range, 0.01 N resolution) clamped to sturdy overhead retort stands, supporting the beam horizontally via knife-edge stirrups at fixed positions x_A = 0.100 m and x_B = 0.900 m.
- Movable rider and concentrated brass load: A calibrated 5.000 N slotted brass mass secured onto a lightweight sliding collar with an index pointer that positions the concentrated load precisely along the beam span.
- Precision spirit level and vertical plumblines: A miniature tubular spirit level placed centrally along the upper beam surface to ensure strict horizontal alignment, confirming that the dynamometer suspension cords remain strictly vertical.
- Hardwood beam length L = 1.000 m; center of gravity x_cg = 0.500 m; uniform self-weight W_bar = 2.000 N.
- Vertical suspension dynamometers at support A (x_A = 0.100 m) and support B (x_B = 0.900 m); span d_AB = 0.800 m.
- Reference concentrated load weight W_load = 5.000 N (mass m ≈ 0.5099 kg); local gravity g = 9.80665 m s⁻².
4. Theory and graph choice
When an extended rigid body is subjected to a system of non-concurrent coplanar forces whose lines of action do not intersect at a common point, static mechanical equilibrium requires that two vector conditions be satisfied simultaneously: the net external vector force must equal zero (translational equilibrium, Σ F = 0), and the net external vector torque about any arbitrary reference axis must vanish identically (rotational equilibrium, Σ τ = 0). In this classical mechanics investigation, a uniform horizontal rigid beam of total length L = 1.000 m and self-weight W_bar = 2.000 N (with center of gravity at x_cg = 0.500 m) is suspended horizontally between two overhead spring dynamometers at support A (x_A = 0.100 m) and support B (x_B = 0.900 m), providing a clear span d_AB = x_B - x_A = 0.800 m. A movable concentrated load of unknown weight W_load = 5.000 N is placed at variable coordinate position x along the beam span. Vertical translational equilibrium requires that the sum of the two upward support reaction forces exactly balances the total downward gravitational load: Σ F_y = R_A + R_B - (W_bar + W_load) = 0, meaning R_A + R_B = 7.000 N at all times. Taking rotational moments about support A eliminates reaction force R_A and yields Σ τ_A = R_B · d_AB - W_bar · (x_cg - x_A) - W_load · (x - x_A) = 0. Solving for the right support reaction R_B reveals a direct linear relationship with respect to the load position x: R_B(x) = (W_load / d_AB) · x + [W_bar · (x_cg - x_A) - W_load · x_A] / d_AB = S · x + b. Here, the theoretical slope S = W_load / d_AB (in N m⁻¹) directly reflects the load weight distributed across the span, while the intercept b reflects the combined geometry and beam tare weight. By measuring reaction force R_B across five distinct load positions x, and performing ordinary least-squares regression, the concentrated load weight is extracted as W_load = S · d_AB = S · 0.800 N, and the beam self-weight W_bar is verified from intercept b, validating the twin pillars of rigid-body statics.
The calculation fits y = mx + b in SI units. The graph may display scaled axes for readability; those display factors are removed before calculating the physical result. The gradient has units N m⁻¹.
5. Method
- Mount the unweighted hardwood beam between the two suspension dynamometers at support coordinates x_A = 0.100 m and x_B = 0.900 m. Adjust the vertical clamp heights on the retort stands until the central spirit level indicates perfect horizontal balance. Record tare readings R_A0 and R_B0 (each nominally 1.00 N).
- Sum the two unloaded dynamometer readings and confirm that R_A0 + R_B0 = W_bar = 2.00 N. Verify that taking moments about support A yields R_B0 · d_AB = W_bar · (x_cg - x_A), proving that the unweighted beam satisfies both equilibrium criteria.
- Attach the sliding rider with load W_load = 5.000 N at initial coordinate x = 0.22 m. Allow all motion to damp out completely, confirm the spirit level remains centered, and record the right dynamometer reaction reading R_B to the nearest 0.01 N.
- Simultaneously read left dynamometer reaction R_A. Verify that the sum of the reactions R_A + R_B equals the total system weight W_total = W_bar + W_load = 2.00 N + 5.00 N = 7.00 N within 0.02 N experimental tolerance.
- Systematically slide the load rider to positions x = 0.34 m, 0.50 m, 0.66 m, and 0.78 m. At each station, check beam horizontality, let transient oscillations settle, and record the corresponding right reaction force R_B.
- Plot right reaction R_B against load position x. Fit an ordinary least-squares regression line to compute slope S and intercept b. Multiply slope S by span d_AB = 0.800 m to determine W_load, and compute beam weight W_bar from intercept b.
6. Observations and calculation table
All observations use fixed model settings. Values are rounded only at the instrument reading stage; further calculations use full precision. Repeated physical trials are not claimed.
| No. | Load position x (m) | Support reaction R_B (N) | x (SI) | y (SI) | Fitted y (SI) | Residual (SI) |
|---|---|---|---|---|---|---|
| 1 | 0.22 | 1.75 | 0.22000 | 1.75000 | 1.75000 | 0.00000 |
| 2 | 0.34 | 2.50 | 0.34000 | 2.50000 | 2.50000 | 0.00000 |
| 3 | 0.50 | 3.50 | 0.50000 | 3.50000 | 3.50000 | 0.00000 |
| 4 | 0.66 | 4.50 | 0.66000 | 4.50000 | 4.50000 | 0.00000 |
| 5 | 0.78 | 5.25 | 0.78000 | 5.25000 | 5.25000 | 0.00000 |
Download the sample observations (CSV) ↓
7. Worked calculation for one observation
8. Graph, gradient and final calculation
Dots show the sample observations. The line is the ordinary least-squares fit.
Σ(x − x̄)² = 0.20800
Σ(x − x̄)(y − ȳ) = 1.30000
m = Σ(x − x̄)(y − ȳ) / Σ(x − x̄)² = 6.25000000
b = ȳ − mx̄ = 0.37500000
Slope S = 6.250000 N m⁻¹; W_load = S × d_AB = 6.250000 × 0.800 m = 5.0000 N
9. Uncertainty and comparison
Percentage difference = |estimate − model value| / |model value| × 100 = 0.0000%. Percentage difference is not experimental uncertainty.
For n = 5 observations, Σresidual² = 0.00000; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 0.00000. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 0.00000.
Propagating only the relevant fit coefficient gives a fit-only standard error of 0.00000 N. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.
Reading resolution of the tubular dynamometers (±0.01 N), beam leveling angle tilt error causing non-vertical suspension, knife-edge bearing contact offset, and wood moisture mass variation contribute uncertainty. Dynamometer spring zero offsets must be calibrated under vertical tare. The regression slope standard error quantifies random load-position reading scatter, but does not capture shared calibration offsets in dynamometer spring constants.
10. Result and discussion
Model comparison value: 5.0000 N
The observations follow the stated model relationship. Check whether the intercept and residual pattern suggest a systematic offset or a missing approximation. Agreement with the model does not independently verify its input constants.
11. Precautions and limitations
- Calibrate and zero both tubular dynamometers in the vertical suspension orientation prior to mounting the beam.
- Ensure the rigid beam remains strictly horizontal using the spirit level so dynamometer suspension strings remain vertical without lateral force components.
- Move the load rider gently along the beam without bumping or jarring to prevent dynamic oscillation and spring calibration shock.
- Verify that the suspension stirrups pivot freely on their knife edges without binding or rubbing against the overhead support stands.
The simulation assumes an idealized perfectly rigid beam without elastic bending deflection, frictionless knife-edge stirrups, and linear Hookean dynamometer springs. Simulated readings reflect a seasoned hardwood beam (tare weight W_bar = 2.000 N) carrying a reference concentrated load W_load = 5.000 N with dynamometer resolution of 0.01 N.
12. Conclusion
The worked example illustrates how to obtain concentrated load weight from measurements, graph variables and a fitted coefficient. The estimate is 5.0000 N. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.