PhysicsSolver360 logoPhysicsSolver360°
Powered by Solver360°

Center of Mass Calculator

Calculate the center of mass position for two point masses on a line

Parameters

kgⓘ
kgⓘ
mⓘ
mⓘ
Show Trail

Controls

xⓘ

Calculated Values

Center of Mass:
2.00;m2.00;m
Total Mass:
5.00;kg5.00;kg
Distance from m₁:
2.00;m2.00;m
Distance from m₂:
3.00;m3.00;m

Examples

3 kg and 2 kg

At x = 0 and x = 5 m.

  • Center of Mass: 2.002.00

Visualization

Center of Mass

The center of mass (COM) is the weighted average position of all mass in a system. For discrete particles: x_COM = (Σ m_i x_i) / (Σ m_i).

For two masses on a line: x_COM = (m₁x₁ + m₂x₂)/(m₁ + m₂). The COM lies between them, closer to the heavier mass.

If no external force acts on a system, the COM moves at constant velocity (or stays at rest). Internal forces (e.g. in collisions) cannot accelerate the COM.

Treat the entire system as a point mass at the COM for analyzing external forces and overall motion. This simplifies rocket motion and collision problems.

For continuous bodies, integrate mass distribution; symmetry often locates COM by inspection.

Key Concepts

  • x_COM = Σm_i x_i / Σm_i — discrete particles
  • COM between two masses on a line
  • Heavier mass pulls COM closer
  • COM velocity constant if F_ext = 0
  • Useful for collisions and system motion
  • 3D: same idea with vector components

Real-World Applications

  • Collision and explosion analysis
  • Balancing objects and stability
  • Sports biomechanics and diving
  • Spacecraft staging and fuel burn
  • Engineering: crane loads and tipping

Explore Further

More mechanics tools

Physics Equations

Center of Mass:
xCOM=m1x1+m2x2m1+m2x_{COM} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}
Total Mass:
M=m1+m2M = m_1 + m_2

Step-by-Step Solution

See how the main results are calculated.

1

Calculate Total Mass

Sum the masses of all particles in the system:

Equation:

M=m1+m2M = m_1 + m_2

Calculation:

M=3+2=5 kgM = 3 + 2 = 5 \text{ kg}

Result:

M=5 kgM = 5 \text{ kg}

Explanation:

For more than two bodies, add all masses: M = Σmᵢ.

2

Compute Weighted Position Sum

Multiply each mass by its coordinate and add:

Equation:

∑mixi=m1x1+m2x2\sum m_i x_i = m_1 x_1 + m_2 x_2

Calculation:

m1x1+m2x2=(3)(0)+(2)(5)m_1 x_1 + m_2 x_2 = (3)(0) + (2)(5)
=0.00+10.00=10.00 kg\cdotpm= 0.00 + 10.00 = 10.00 \text{ kg·m}

Result:

10.00 \text{ kg·m}

Explanation:

Heavier masses contribute more to the COM position—this is a mass-weighted average, not a simple midpoint.

3

Find Center of Mass

Divide the weighted sum by total mass:

Equation:

xCOM=m1x1+m2x2m1+m2x_{COM} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}

Calculation:

xCOM=10.005=2.00 mx_{COM} = \frac{10.00}{5} = 2.00 \text{ m}

Result:

xCOM=2.00 mx_{COM} = 2.00 \text{ m}

Explanation:

COM lies 2.00 m from m₁ and 3.00 m from m₂. It is always between the two masses on a line.

Frequently Asked Questions (FAQ)

Is COM always between the masses?

For two point masses on a line, yes—the COM always lies between them, closer to the larger mass.

Can internal forces move the COM?

No. Only external forces can change the velocity of the center of mass.

What if one mass is much larger?

The COM is very close to the heavier mass: x_COM ≈ x_heavy.

Is COM the same as geometric center?

Only for uniform symmetric objects; for unequal point masses they differ.

Practice MCQs

  1. If m₁ = m₂, COM is at:
  2. Doubling m₁ (positions fixed) moves COM:
  3. Total mass in COM formula is:

Related Calculators

These tools connect to the same physics concepts used in this calculator.

Projectile MotionCalculate and visualize the path of objects under the influence of gravity.Newton's LawsApply the three laws of motion to solve force and acceleration problems.Circular MotionAnalyze objects moving in circular paths with centripetal acceleration.Conservation of MomentumSolve problems involving collisions and momentum conservation.Error PropagationCombine measurement uncertainties for sums, products, quotients, and powers in experimental formulas.Related Measurement & Uncertainty topic.Ideal Gas LawCalculate pressure, volume, temperature, and moles for ideal gases.Related Thermodynamics topic.Bernoulli's PrincipleCalculate fluid velocities, pressures, and flow rates using Bernoulli's equation.Related Fluid Mechanics topic.