Center of Mass Calculus Calculator

Calculator Input

Example Data Table

Formula Used

Variable Density Rod

Total mass is M = integral from a to b of rho(x) dx. The moment is My = integral from a to b of x rho(x) dx. The center of mass is x-bar = My / M.

Lamina Between Curves

For a region between y = f(x) and y = g(x), mass is M = rho integral from a to b of [f(x) - g(x)] dx. The x-coordinate is x-bar = My / M, where My = rho integral from a to b of x[f(x) - g(x)] dx. The y-coordinate is y-bar = Mx / M, where Mx = rho over 2 times integral from a to b of [f(x)^2 - g(x)^2] dx.

Parametric Wire

For x(t), y(t), and density lambda(t), use ds = square root of [(dx/dt)^2 + (dy/dt)^2] dt. Mass is M = integral lambda(t) ds. Coordinates are x-bar = integral x(t) lambda(t) ds / M and y-bar = integral y(t) lambda(t) ds / M.

How to Use This Calculator

Select the model type first. Enter the required functions using x or t. Add lower and upper bounds. Choose enough Simpson slices for accuracy. Press the calculate button. The result appears above the form and below the header. Use the CSV or PDF button to save the computed mass, moments, and coordinates.

Center of Mass Calculus Guide

Why Center of Mass Matters

Center of mass calculus links geometry, density, and balance. It replaces guessing with integration. A shape can be simple, curved, hollow, or uneven. The method still follows the same idea. Add every tiny piece of mass. Add every tiny turning effect. Divide the moment by the total mass.

Rod Problems

For a thin rod, each small piece is treated as dm. The density function rho of x tells how heavy that tiny piece is. The total mass is the integral of rho. The moment about the origin is the integral of x times rho. The quotient gives the balancing point on the line.

Lamina Problems

For a flat lamina, calculus uses area strips. When the region lies between a top curve and a lower curve, each strip has height f of x minus g of x. The mass equals density times the area integral. The x moment uses x times strip area. The y moment uses one half of the difference of squared curve values. These formulas give the centroid when density is constant.

Curved Wire Problems

For a curved wire, the small mass depends on arc length. The differential length comes from the derivatives of the parametric coordinates. A density function may also change along the parameter. This makes the model useful for cables, rails, and paths.

Numerical Method

Numerical integration is helpful when formulas are long. Simpson’s rule estimates each integral by sampling many points. More slices usually improve accuracy. Smooth functions work best. Sharp corners or breaks need careful bounds.

Export and Review

This calculator is designed for study and checking. It shows mass, moments, coordinates, and the method used. It also creates a sample table. The export buttons help save work. CSV is useful for spreadsheets. PDF is useful for reports and assignments.

Accuracy Tips

Always review the entered functions. Use radians for trigonometric expressions. Keep upper bounds larger than lower bounds. Include units if needed. A correct setup matters more than a decimal result.

Advanced Checks

Advanced users can compare several models. A rod may use nonuniform density. A lamina may use signed curve gaps. A wire may use parametric speed. These choices describe real objects better. They also reveal common errors. If mass becomes zero or negative, check the bounds, curves, and density before trusting results during serious math work.

FAQs