Enter Your Values
Use rod mode for a weighted line, or lamina mode for a region between two curves.
Plotly Graph
The chart shows the active region or density curve and marks the computed centroid.
Integral Formulas Used
Rod on the x-axis
For a rod with density rho(x) on [a, b], the mass is M = integral from a to b of rho(x) dx. The centroid is x-bar = (1/M) integral from a to b of x rho(x) dx, with y-bar = 0.
The second moment about the y-axis is Iy = integral from a to b of x2 rho(x) dx.
Lamina between two curves
For a region between upper(x) and lower(x) with areal density delta(x), the mass is M = integral from a to b of delta(x)[upper(x)-lower(x)] dx.
The centroid coordinates are x-bar = (1/M) integral from a to b of x delta(x)[upper-lower] dx and y-bar = (1/M) integral from a to b of delta(x)[upper(x)2 - lower(x)2] / 2 dx.
This file evaluates the integrals numerically with Simpson's rule for fast and stable estimates.
How to Use This Calculator
- Choose Rod for a weighted line, or Lamina for a region between two x-based curves.
- Enter the lower and upper integration limits.
- Provide density coefficients for rho(x).
- When using lamina mode, also enter lower and upper curve coefficients.
- Pick the number of intervals, graph points, decimal places, and a unit label.
- Press Calculate center of mass to show the result above the form.
- Use the CSV and PDF buttons to export your result summary or the worked example table.
Example Data Table
This sample uses lamina mode with lower(x) = 1 + 0.1x, upper(x) = 5 - 0.2x^2, and density rho(x) = 1.5 + 0.2x.
| x | lower(x) | upper(x) | density rho(x) | strip height | approx strip mass for width 0.5 |
|---|---|---|---|---|---|
| 0.00 | 1.000 | 5.000 | 1.500 | 4.000 | 3.000 |
| 1.00 | 1.100 | 4.800 | 1.700 | 3.700 | 3.145 |
| 2.00 | 1.200 | 4.200 | 1.900 | 3.000 | 2.850 |
| 3.00 | 1.300 | 3.200 | 2.100 | 1.900 | 1.995 |
| 4.00 | 1.400 | 1.800 | 2.300 | 0.400 | 0.460 |
Frequently Asked Questions
1. What does this calculator compute?
It computes total mass, centroid coordinates, first moments, and second moments. Rod mode returns a one-dimensional centroid. Lamina mode returns both x-bar and y-bar for a region between two curves.
2. Which shapes are supported?
This version supports a rod on the x-axis and a lamina bounded by two polynomial curves in x. These two models cover many textbook center-of-mass exercises and practice problems.
3. Why does the interval count need to be even?
The file uses Simpson's rule for numerical integration. Simpson's rule splits the interval into pairs of subintervals, so an even interval count is required for the method to work correctly.
4. Can I enter negative density coefficients?
Yes, the calculator allows them mathematically. However, negative density is usually not physical. Signed density can make the total mass very small, which may produce unstable or nonphysical centroid values.
5. What happens when upper(x) is below lower(x)?
Those strips are clipped to zero height so the area stays meaningful. A warning appears when the file detects this case across the selected interval.
6. Are the answers exact or approximate?
The calculator uses numerical integration, so the answers are approximations. For smooth polynomial inputs and a reasonable interval count, the estimates are usually very accurate for study and checking work.
7. How should I choose units?
Keep every input in a consistent system. If x is measured in centimeters, then x-bar and y-bar will also be in centimeters. Mass depends on the density units you assume.
8. What does the graph show?
In rod mode, the graph shows the density curve and the centroid location on the x-axis. In lamina mode, it draws the region between curves and places a marker at the computed centroid.