Hardwood meter rule
A seasoned, uniform hardwood meter rule of total length 100.0 cm, having a rectangular cross-section, sharp millimeter graduations along both top and bottom edges, and a verified center of gravity at x_cg = 50.00 cm.
Physics / Moments & center of gravity
FREE TO EXPLOREEXPERIMENT 16 · PHYSICS LABORATORY
Principle of moments and rotational equilibrium of parallel forces
Verify the principle of moments for parallel coplanar forces acting on a rigid beam, demonstrate that rotational equilibrium requires the algebraic sum of clockwise and counter-clockwise torques to vanish about an off-center pivot axis, and determine the unknown mass and center of gravity of a uniform meter rule using ordinary least-squares regression. Explore a virtual apparatus, collect readings and analyse a graph.
CHANGE · OBSERVE · MEASURE
Your virtual workbench.
BALANCING DISTANCE D_M—
MEASUREMENT STATUS
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.
Simulation assumptions and limits
The model assumes a perfectly rigid, homogeneous beam, zero knife-edge contact bearing width, frictionless knife-edge pivot, and negligible air buoyancy on the extended arms. Simulated readings reflect a seasoned hardwood meter rule (reference mass M_rule = 0.1200 kg = 120.0 g) with millimeter scale reading resolution of 0.5 mm.
Know your apparatus.
Wedge fulcrum and heavy retort stand
A hardened tool-steel knife-edge prism clamped horizontally atop a heavy cast-iron laboratory stand fitted with leveling foot-screws, providing a frictionless line contact at the fixed graduation mark x_0 = 35.0 cm.
Slotted brass masses and suspension loop
A calibrated set of precision brass masses (60.0 g to 150.0 g) suspended from a lightweight wire stirrup with a fine nylon loop that slides smoothly along the meter rule arm to achieve exact rotational balance.
Miniature spirit level and analytical balance
A tubular glass spirit level placed along the center axis of the meter rule to indicate true horizontal alignment, and an electronic laboratory balance (0.01 g precision) used to determine the independent reference comparison mass.
FOLLOW THE METHOD
A reason behind every step.
Verify the principle of moments for parallel coplanar forces acting on a rigid beam, demonstrate that rotational equilibrium requires the algebraic sum of clockwise and counter-clockwise torques to vanish about an off-center pivot axis, and determine the unknown mass and center of gravity of a uniform meter rule using ordinary least-squares regression. Suggested independent settings: 60.0, 80.0, 100.0, 120.0, 150.0 g.
Determine the rule center of gravity
Place the unweighted hardwood meter rule horizontally across the steel knife-edge fulcrum and adjust its position until it rocks gently and balances in a steady horizontal attitude without tipping. Note the graduation mark directly above the knife edge as the center of gravity x_cg (nominally 50.0 cm for a homogeneous rule).
Why this step matters
Identifying the true center of gravity confirms that the rule's distributed gravitational load can be mathematically replaced by a single concentrated vertical downward force M_rule · g acting precisely through this line of symmetry.
Set the off-center fulcrum position
Shift the knife-edge fulcrum stand and position the meter rule so that graduation mark x_0 = 35.0 cm rests securely upon the prism edge. Confirm that the rule tilts downward on the longer arm due to the unbalanced clockwise gravitational torque exerted by the rule's self-weight.
Why this step matters
Offsetting the fulcrum away from the center of gravity establishes a known, constant lever arm d_rule = x_cg - x_0 = 15.0 cm (0.150 m), generating the reference gravitational torque that will be counteracted by suspended test loads.
Attach initial balancing mass and achieve equilibrium
Load a total mass m = 60.0 g onto the suspension stirrup and slide its fine thread loop along the shorter arm of the rule (between 0.0 cm and 35.0 cm). Slowly shift the loop until the meter rule levels out horizontally, confirmed by the central spirit level bubble resting centered between its index lines.
Why this step matters
The suspended mass exerts a downward tension force m · g that produces a counter-clockwise torque about the pivot. Equilibrium occurs when the counter-clockwise torque exactly balances the clockwise gravitational torque of the beam.
Record balancing graduation and calculate moment arm
Read the exact position of the suspension thread on the rule's millimeter scale, using an eye position perpendicular to the rule to eliminate optical parallax error. Calculate the balancing moment arm distance d_m = x_0 - x_m. Repeat the leveling check three times to verify mechanical repeatability.
Why this step matters
Measuring the suspension coordinate x_m enables direct computation of the perpendicular distance d_m to the pivot axis, which forms the dependent variable in the moment equilibrium equation.
Repeat across incremental mass settings
Systematically increase the suspended load to m = 80.0 g, 100.0 g, 120.0 g, and 150.0 g. At each mass increment, slide the suspension stirrup to find the new horizontal balance position, verify with the spirit level, and record the corresponding balancing distance d_m.
Why this step matters
Exploring multiple load settings distributes observations over a broad lever arm range (from 12.0 cm to 30.0 cm), minimizing parameter leverage and testing the inverse linear relationship across wide dynamic variations.
Transform variables and perform linear regression
Convert all masses to kilograms and compute reciprocal mass 1/m (in kg⁻¹). Convert balancing distances to meters. Plot d_m against 1/m, fit an ordinary least-squares regression line, compute the slope S, and divide by d_rule = 0.150 m to deduce the meter rule mass M_rule.
Why this step matters
Linear regression separates zero-scale offsets and thread tare errors into the intercept b, yielding an unbiased estimate of the slope S with associated statistical confidence intervals.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into a result.
Points are your observations. The fit uses SI x and y values before display scaling.
LINEAR FIT · y = mx + b
d_m = (M_rule · d_rule) · (1/m) + b
| # | Balancing mass m (g) | Balancing distance d_m (cm) | Reciprocal mass 1/m (kg⁻¹) | Balancing distance d_m (m) | Remove |
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No readings yet. Take a virtual measurement and record it.
Calculate it yourself.
CHECK YOUR UNDERSTANDING