Center of Mass Calculator
Center of Mass Calculator computes weighted averages for points or distributions, showing steps, examples for physics problems.
Equation Preview
Helping Notes
- Only required inputs: each point’s mass and coordinates (x, y, z by dimension). :contentReference[oaicite:2]{index=2}
- Formula (vector form): \(\mathbf{r}_{\text{CM}}=\frac{1}{M}\sum m_i\mathbf{r}_i\), with \(M=\sum m_i\). :contentReference[oaicite:3]{index=3}
- 2D: \(x_{CM}=\frac{\sum m_ix_i}{M}\), \(y_{CM}=\frac{\sum m_iy_i}{M}\). 3D adds \(z_{CM}=\frac{\sum m_iz_i}{M}\). :contentReference[oaicite:4]{index=4}
- Leave an entire mass row blank to ignore it. Masses must be positive.
Results
Center of Mass
Totals
Error
Steps
What is Center of Mass Calculator?
A Center of Mass Calculator finds the balance point of a system by averaging positions weighted by mass. For discrete particles with masses \(m_i\) at position vectors \(\mathbf r_i\), the center of mass (CoM) is the unique point where total torque from gravity vanishes. In practice it is a mass‑weighted mean, reducing to the geometric centroid when mass is uniformly distributed. The core formulas are:
For continuous bodies with density functions, integrals replace sums:
About the Center of Mass Calculator
The calculator supports 1D (line), 2D (plane), and 3D inputs. Provide discrete points with masses or pick density‑based modes (uniform or variable) for rods, laminae, and solids. It computes total mass, center‑of‑mass coordinates, and shows a clear step table: partial products \(m_i x_i\), \(m_i y_i\), \(m_i z_i\); running sums; and final division by \(M\). For continuous cases, it displays the corresponding integrals and evaluates them symbolically when possible, otherwise numerically. Units are preserved throughout—enter consistent length and mass units to obtain interpretable results.
How to Use this Center of Mass Calculator
- Select the problem type: discrete points, rod (1D), lamina (2D), or solid (3D).
- Enter coordinates and masses (or density function and region/interval bounds).
- Press calculate to compute \(M\) and \(\mathbf r_{\mathrm{cm}}\).
- Review the step‑by‑step table or integral evaluation presented in responsive formula blocks.
- Copy the final CoM and intermediate steps for reports, labs, or homework.
Examples
Example 1: Two masses on a line
\(m_1=2\,\text{kg}\) at \(x_1=0\,\text{m}\); \(m_2=3\,\text{kg}\) at \(x_2=4\,\text{m}\).
Example 2: Two points in the plane
\(m_1=3\) at \((-1,2)\), \(m_2=1\) at \((4,0)\).
Example 3: Rod with variable density
Rod on \([0,1]\) with \(\lambda(x)=2x\). Then
FAQs
Is center of mass the same as centroid?
For uniform density it coincides with the centroid; with varying density, the center of mass weights by \(\rho\).
Do coordinates depend on the origin?
Numerical values change with origin choice, but the physical location is fixed in space relative to the body.
Can I mix units (e.g., grams and kilograms)?
Avoid mixing. Convert all masses and lengths to consistent units to get meaningful coordinates.
Does negative mass make sense here?
Physical mass is nonnegative. Mathematically, negative weights can represent corrections, but interpret with caution.
How is center of gravity different?
In a uniform gravitational field, center of gravity aligns with center of mass; in nonuniform fields they can differ slightly.