Center of Mass from Density Calculator

Enter the polynomial coefficients and the rod's left and right endpoints. The calculator evaluates the total mass integral and the first moment integral analytically, then divides the first moment by the mass. It returns both integrals alongside the center of mass, making the result easy to inspect in calculus, mechanics, engineering, or automated coursework workflows.

Describe the rod with a polynomial density

Represent the linear density as a polynomial in position x and enter its coefficients from the highest power down to the constant term. For example, [2, 1] means rho(x) = 2x + 1, while [3, 0, -2] means rho(x) = 3x squared - 2. Keep explicit zeros wherever a power is missing so the degree and every term remain unambiguous. The lower and upper bounds locate the two ends of the rod on the same coordinate axis used by the density formula. The upper bound must be strictly greater than the lower bound. Coefficients and bounds must be finite numbers, and the resulting total mass must be positive. Density normally has units of mass per length, such as kilograms per meter. If x is measured in meters and density is kilograms per meter, the mass result is in kilograms, the first moment is in kilogram-meters, and the center of mass is in meters. The calculator does not attach or convert units, so use one consistent system throughout. A physical linear-density model should also be nonnegative across the rod; confirm that property when choosing the polynomial.

How the mass and first moment integrals are evaluated

For a rod extending from a to b with linear density rho(x), total mass is the definite integral of rho(x) from a to b. The first moment about the coordinate origin is the definite integral of x times rho(x) over the same interval. This calculator integrates each polynomial term directly with the power rule. If a density term is c times x to the n, its contribution to mass is c times x to the n plus one divided by n plus one, evaluated at both endpoints. Multiplying density by x raises the power by one before the same rule is applied to the first moment. The center of mass is then x-bar = first moment divided by total mass. No sampling grid, iterative solver, symbolic parser, or numerical quadrature is involved, so there is no step size to choose and no convergence setting to tune. The response exposes total_mass and first_moment instead of returning only their ratio. Those intermediate values help you verify an antiderivative, find a misplaced coefficient, and preserve dimensional reasoning. Arithmetic uses finite JavaScript numbers and stable output normalization; calculations that overflow the supported numeric range are rejected rather than emitting unusable infinity values.

Interpret the balance point and check the model

The center_of_mass value is the coordinate at which the rod would balance under the supplied density model. For a constant positive density on an interval, it equals the midpoint because equal lengths contain equal mass. When density increases toward the right, the center generally shifts right of the geometric midpoint; when density is heavier toward the left, it shifts left. These checks are useful before accepting a result. For a physically valid nonnegative density with positive mass, the center should lie between the endpoints. An unexpected position often indicates coefficients entered in ascending rather than descending order, inconsistent coordinates, or a density polynomial that becomes negative somewhere on the interval. The service requires a positive mass integral but does not prove that the polynomial is nonnegative at every point, so responsibility for the physical suitability of the density function remains with the caller. This distinction also permits calculus exercises involving an already-approved polynomial without hiding the actual integral computation. Use the returned degree and bounds to confirm what was evaluated, retain the unrounded response for later calculations, and round only when presenting the final coordinate at the precision justified by the original measurements.