Use both area integrals for a nonuniform lamina
A planar center of mass requires two first moments over the same area and the same total mass. Our rectangle spans 0 to 2 m east and 0 to 1 m north, with density σ = σ₀(1 + ax + by). The supplied a and b are nonnegative gradients in inverse metres; total mass is fixed at 4 kg.
Learning outcome: Reconstruct each double integral and compare a single-gradient case with the uniform limit. Record density normalization and both first moments.
Starting recap
Integrate signed force and normalized mass moments. Specify the system before conserving momentum, and independently test energy and observation constraints.
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At a = 0 and b = 1 m⁻¹, xcm remains 1 m by east-west symmetry and ycm = 5/9 m. Uniform density restores (1, 1/2) m.
Open the experimentPredict
At a = 0 and b = 1 m⁻¹, which center is correct?
Experiment
Reconstruct each double integral and compare a single-gradient case with the uniform limit. Record density normalization and both first moments.
Integrate both first moments of a planar density
Worked teaching model · synthetic values · Rectangular lamina0≤x≤2 m,0≤y≤1 m with density rho0(1+a x+b y); rho0 normalized to total mass4 kg. Fixed flat geometry, nonnegative supplied density, no actual material fabrication or tomography.
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Lamina boundary and mass center
Spatial axes use the same pixels per physical unit, preserving geometry. 1: rectangular boundary (solid).2: center cross (dashed). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Lamina boundary and mass center
rectangular boundary
| east position (m) | north position (m) |
|---|---|
| 0 | 0 |
| 2 | 0 |
| 2 | 1 |
| 0 | 1 |
| 0 | 0 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
center cross
| east position (m) | north position (m) |
|---|---|
| 1.098148 | 0.5185185 |
| 1.198148 | 0.5185185 |
| 1.148148 | 0.5185185 |
| 1.148148 | 0.4685185 |
| 1.148148 | 0.5685185 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Normalized base area density | 0.8888889 | kg/m² |
| Total mass | 4 | kg |
| East center coordinate | 1.148148 | m |
| North center coordinate | 0.5185185 | m |
| East first moment | 4.592593 | kg m |
| North first moment | 2.074074 | kg m |
Both first moments require integration over the entire lamina, not just one representative edge.
Integrate rho0(1+a x+b y) over x0to2,y0to1. Mass/rho0=2+2a+b; east moment/rho0=2+8a/3+b; north moment/rho0=1+a+2b/3. Divide moments by the same normalized total mass. Uniform density a=b=0 gives(1,.5), and both normalized gradients remain nonnegative throughout the supplied rectangle.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Both first moments require integration over the entire lamina, not just one representative edge.
Integrate rho0(1+a x+b y) over x0to2,y0to1. Mass/rho0=2+2a+b; east moment/rho0=2+8a/3+b; north moment/rho0=1+a+2b/3. Divide moments by the same normalized total mass. Uniform density a=b=0 gives(1,.5), and both normalized gradients remain nonnegative throughout the supplied rectangle.
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What symmetry survives when only the north gradient changes?
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