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. Explore a virtual apparatus, collect readings and analyse a graph.
Ready for a virtual observation.
Adjust the apparatus, then take a reading.
Readings are generated by an ideal model. The diagram is schematic and is not to scale.
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.
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.
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.
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.
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.
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. Suggested independent settings: 20.0, 30.0, 40.0, 50.0, 60.0 °.
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.
Any tilt in the table surface introduces an unmodeled gravitational component acting parallel to the plate, causing the central ring to slide downhill and biasing the force equilibrium measurements.
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.
Symmetric angular placement forces the lateral orthogonal components to cancel exactly (+M_0 g sin θ - M_0 g sin θ = 0), aligning the net vector resultant R purely along the 0° Cartesian axis.
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).
Static equilibrium demands an equilibrant vector E that is collinear, equal in magnitude, and oppositely directed to the resultant vector R (E = -R), which corresponds strictly to 180° on the circle.
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.
Free concentric flotation of the ring without touching the centering pin proves that the net vector sum of the three concurrent tension forces equals zero (Σ F = 0).
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.
Testing across five distinct angles spans a wide range of cosine values (0.500 to 0.940) and equilibrant masses (150 g to 282 g), providing robust statistical leverage for linear regression.
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.
Linear regression averages out random ring-centering reading errors and isolates any minor cord tare offsets into the intercept b, yielding an unbiased estimate of M_0 with rigorous confidence intervals.
Points are your observations. The fit uses SI x and y values before display scaling.
| # | Symmetric half-angle θ (°) | Equilibrant mass m_E (g) | cos(θ) | Equilibrant mass m_E (kg) | Remove |
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