Force table & vector addition lab report

1. Title and aim

Vector addition and equilibrium of concurrent coplanar forces on a force table. Investigate the static mechanical equilibrium of concurrent coplanar forces using a precision circular force table, verify the parallelogram and polygon laws of vector addition by resolving forces into orthogonal Cartesian components, and determine the unknown magnitude of equal symmetric load forces 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 component load mass of 0.1500 kg, differing by 0.002% 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

• Circular 360° graduated force table: A precision-machined circular aluminum table graduated in single degrees from 0° to 360° along its outer raised rim, supported on a heavy cast-iron tripod stand fitted with three adjustable leveling foot-screws and a central removable centering pin.
• Low-friction clamp-on pulleys: Three miniature ball-bearing pulleys equipped with knurled thumbscrew clamps that secure rigidly to the table rim, designed so the pulley groove plane aligns strictly with the table radius toward the center pin.
• Central floating ring and radial cords: A lightweight aluminum ring of negligible mass connected to three non-stretch braided cords that radiate outward across the pulleys, maintaining concentric floating alignment around the centering pin during equilibrium.
• Precision slotted brass weights and hangers: A set of standardized slotted brass weights (1.0 g to 100.0 g) and lightweight 50.0 g weight hangers, calibrated on an analytical laboratory balance (0.01 g resolution) to apply precise gravitational tensions.
• Machined aluminum circular force table (diameter D = 40.0 cm) graduated in 1° divisions from 0° to 360°.
• Symmetric applied load masses F_1 = F_2 = M_0 · g with reference mass M_0 = 0.1500 kg (150.0 g).
• Equilibrant pulley clamped at fixed angle 180.0°; local acceleration of gravity g = 9.80665 m s⁻².

4. Theory and graph choice

In classical mechanics, forces are vector quantities possessing both magnitude and direction, obeying the axioms of vector addition rather than simple scalar arithmetic. When multiple coplanar forces act concurrently through a single common point of application, static translational equilibrium requires that their vector sum vanishes identically: Σ F = F_1 + F_2 + ... + F_n = 0. In this laboratory investigation, concurrent force equilibrium is studied using a precision circular force table equipped with a 360° graduated rim, radial low-friction pulleys, and a central ring floating around a vertical centering pin. Two known equal forces F_1 and F_2, generated by suspended standard masses M_0 = 150.0 g (0.1500 kg), are aligned symmetrically at equal half-angles +θ and -θ relative to the 0° Cartesian x-axis. Resolving each force into orthogonal components yields F_{1x} = M_0 · g · cos(θ), F_{1y} = M_0 · g · sin(θ), F_{2x} = M_0 · g · cos(θ), and F_{2y} = -M_0 · g · sin(θ). Due to the angular symmetry about the x-axis, the lateral vertical components cancel identically: Σ F_y = F_{1y} + F_{2y} = 0. The resultant force vector R acts entirely along the positive x-axis (0°) with magnitude R = R_x = F_{1x} + F_{2x} = 2 M_0 · g · cos(θ). To establish complete equilibrium, a third pulley carrying an equilibrant mass m_E is clamped diametrically opposite at 180°. By definition, the equilibrant force E must be equal in magnitude and opposite in direction to the resultant: E = -R, so E = m_E · g = 2 M_0 · g · cos(θ). Dividing through by the gravitational acceleration g yields the direct linear relationship m_E = (2 M_0) · cos(θ). By measuring the equilibrant mass m_E across five distinct half-angles θ, and plotting m_E against cos(θ), the resulting straight-line slope S = 2 M_0 allows the component mass M_0 = S / 2 to be deduced with high statistical precision, confirming the vector addition law and the validity of orthogonal component resolution.

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 kg.

5. Method

1. Place a circular bullseye spirit level directly onto the center of the force table disk. Adjust the three leveling screws on the tripod base until the spirit level bubble is centered within the target index circle. Rotate the level through 90° to verify that the table is horizontal in all directions.
2. Clamp the first pulley at angle +θ (e.g. 20.0°) and the second pulley at symmetric angle 360° - θ (e.g. 340.0°). Thread two radial cords from the central ring over these pulleys and attach identical total masses M_0 = 150.0 g (including the weight hangers) to both cords.
3. Clamp the third pulley at exactly 180.0° on the table rim, diametrically opposite the 0° line of action of the resultant force. Pass the third cord over this pulley and suspend an empty weight hanger (50.0 g tare).
4. Carefully add slotted brass weights to the equilibrant hanger until the central ring floats freely and concentrically around the centering pin without contacting it. Tap the rim of the table gently with a pencil to eliminate static bearing friction in the pulley axles.
5. Record the total equilibrant mass m_E (weights plus hanger) to the nearest 0.1 g. Systematically alter the symmetric half-angle setting to θ = 30.0°, 40.0°, 50.0°, and 60.0°. At each setting, adjust the equilibrant mass m_E until the ring is perfectly centered, and record the new balance mass.
6. Calculate the independent variable x = cos(θ) for each setting. Convert m_E to kilograms (y = m_E / 1000). Plot y against x, fit an ordinary least-squares regression line, determine the slope S, and compute the estimated component mass M_0 = S / 2.

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.

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.

9. Uncertainty and comparison

Percentage difference = |estimate − model value| / |model value| × 100 = 0.0023%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 7.6141e-10; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 4.5343e-05. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 3.4431e-05.

Propagating only the relevant fit coefficient gives a fit-only standard error of 2.2672e-05 kg. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Angular scale resolution of the circular table (±0.25°), pulley axle bearing friction and string-pulley groove contact resistance, knot stretching and non-coaxial central ring alignment, and slotted mass manufacturing tolerances contribute uncertainty. Tapping the table breaks static friction dead-bands. The regression slope standard error quantifies random ring-centering scatter, but excludes shared systematic zero-degree alignment errors.

10. Result and discussion

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

• Level the circular force table precisely using a bullseye spirit level before clamping any pulleys.
• Ensure all cords radiate strictly along true table radii and pass smoothly over the center of each pulley groove without rubbing against pulley brackets.
• Tap the table rim gently before recording each equilibrant mass to break static axle friction and allow the ring to float freely.
• Keep the central centering pin in place while altering weights to prevent the ring from flying off, but check equilibrium with the ring floating concentrically without touching the pin.

The simulation assumes idealized frictionless pulleys, massless cords, an infinitely thin centering pin, and perfectly concurrent lines of action through the mathematical center. Simulated readings reflect symmetric applied loads of reference mass M_0 = 0.1500 kg (150.0 g) with slotted weight set resolution of 0.1 g.

12. Conclusion

The worked example illustrates how to obtain component load mass from measurements, graph variables and a fitted coefficient. The estimate is 0.1500 kg. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.