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Category Physics & Science

Mass

Presets:

#	Mass	X	Y	Action

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About

Miscalculating the center of mass of a system leads to structural imbalance, failed load distribution, and incorrect torque analysis. This tool computes the exact center of mass coordinates (x_cm, y_cm) for a discrete system of up to 50 point masses in two dimensions using the standard weighted-average formulation from classical mechanics. All mass values must be strictly positive. The calculation assumes point masses in a uniform gravitational field, meaning center of mass and center of gravity coincide. It does not account for continuous mass distributions or relativistic effects.

Accurate center of mass determination is critical in aerospace engineering (satellite attitude control), robotics (stability analysis), civil engineering (load path verification), and game physics (rigid body simulation). The tool provides an interactive canvas where you can place, drag, and edit masses visually. Pro tip: for composite bodies, decompose into sub-shapes, compute each centroid separately, then use this calculator with the sub-shape masses and centroid positions as inputs.

Formulas

The center of mass for a system of n discrete point masses in two dimensions is computed as:

x_cm = n∑i=1 m_i ⋅ x_in∑i=1 m_i

y_cm = n∑i=1 m_i ⋅ y_in∑i=1 m_i

Where m_i is the mass of the i-th particle, x_i and y_i are its Cartesian coordinates, and n is the total number of point masses. The denominator M = n∑i=1 m_i represents the total mass of the system. Each coordinate of the center of mass is the mass-weighted average of the corresponding coordinates of all particles. The result is a single point (x_cm, y_cm) that behaves as if the entire system mass M were concentrated there when analyzing translational motion.

Reference Data

Shape / Configuration	CoM Location (x_cm, y_cm)	Notes
Uniform Rod (length L)	L2 from one end	Along the axis of the rod
Uniform Rectangle (a × b)	(a2, b2)	Geometric center
Uniform Triangle (vertices known)	x₁ + x₂ + x₃3, y₁ + y₂ + y₃3	Centroid at medians intersection
Uniform Circular Disk (radius R)	Center of the circle	By symmetry
Uniform Semicircle (radius R)	(0, 4R3π)	≈ 0.4244R from diameter
Uniform Quarter Circle (radius R)	(4R3π, 4R3π)	Both axes identical by symmetry
Uniform Solid Hemisphere (radius R)	3R8 from flat face	3D result, along symmetry axis
Uniform Solid Cone (height h)	h4 from base	Along central axis
Hollow Hemisphere Shell (radius R)	R2 from flat face	Thin shell assumption
Two Equal Masses	Midpoint between them	Independent of mass magnitude
Two Unequal Masses (m₁, m₂)	Closer to heavier mass	Divides distance in ratio m₂:m₁
L-shaped Composite (equal arms)	Inside the L bend region	Decompose into two rectangles
T-shaped Composite	Along axis of symmetry	Decompose into flange + web
Ring / Annulus	Geometric center	By rotational symmetry
Uniform Parallelogram	Diagonal intersection	Same as rectangle method
Earth-Moon System	≈ 4670 km from Earth center	Inside Earth (radius 6371 km)
Uniform Trapezoid (a, b, h)	h(2b + a)3(a + b) from base a	Along height axis
Uniform Sector (angle θ, radius R)	2R sin(θ/2)3(θ/2) from center	θ in radians
Two-body (general)	m₁r₁ + m₂r₂m₁ + m₂	Fundamental two-body formula

Frequently Asked Questions

The center of mass shifts strongly toward the dominant mass. In the limiting case where one mass m₁ → ∞ relative to all others, the CoM converges to (x₁, y₁). This is why the Earth-Moon system's CoM lies inside Earth despite the Moon's large distance.

Yes. For concave or hollow objects (a ring, an L-shaped bracket, a boomerang), the center of mass often lies in empty space where no material exists. This is physically valid - the CoM is a mathematical point, not necessarily a material point.

In a uniform gravitational field (constant g), center of mass and center of gravity are identical. They diverge only in non-uniform fields, such as very tall structures in planetary-scale gravity gradients. For objects smaller than approximately 100 km on Earth, the difference is negligible.

Decompose the body into simpler sub-shapes (rectangles, triangles, circles). Compute each sub-shape's centroid and area (or volume times density for mass). Then enter each sub-shape as a point mass at its centroid location in this calculator. The composite CoM formula is mathematically identical to the discrete point-mass formula.

The CoM coordinates depend on your chosen origin and axis directions, but the physical location of the CoM point is invariant. Shifting the origin by (Δx, Δy) shifts the computed CoM by the same offset. Rotating axes transforms coordinates accordingly. The relative position of the CoM within the mass distribution remains constant.

Yes, this is a standard engineering technique. To compute the CoM of a plate with a circular hole, enter the full plate as a positive mass at its centroid, and the hole as a negative mass at the hole's centroid. The formula handles negative masses algebraically. Ensure the total mass M remains positive to avoid a division-by-zero or sign inversion.