Find the balance point of a system of particles in 2D space
Center of Mass (X, Y)
Total Mass
📐 Formulas: X_cm = Σ(m_i × x_i) / Σm_i • Y_cm = Σ(m_i × y_i) / Σm_i
The center of mass (COM) is the weighted average position of all mass in a system—the unique point where the system balances perfectly in a uniform gravitational field.
Physical meaning:
If you could support the system at just the COM, it would remain balanced without rotating. All the mass "averages out" to this single point. The COM moves as if all external forces were applied there.
🚗 Vehicle Stability
Lower COM = better stability. SUVs have higher rollover risk due to elevated COM.
🚀 Rocket Design
COM shifts as fuel burns, affecting flight stability and control.
🏗️ Structural Engineering
Load distribution in beams, trusses, and foundations depends on COM calculations.
System: Particle 1: 10 kg at (0, 0), Particle 2: 5 kg at (10, 0)
Step 1: Total Mass = 10 + 5 = 15 kg
Step 2: Weighted X = (10 × 0) + (5 × 10) = 0 + 50 = 50
Step 3: Weighted Y = (10 × 0) + (5 × 0) = 0
Step 4: X_cm = 50 / 15 = 3.33 m
Step 5: Y_cm = 0 / 15 = 0 m
✓ Final Answer: Center of mass is at (3.33, 0)
The COM is closer to the heavier 10 kg particle, as expected. It's located 1/3 of the distance from the 10 kg particle toward the 5 kg particle.
Yes! The COM of a hollow object can be in empty space. Classic example: a donut (torus) has its COM in the center hole where there's no material. Similarly, a horseshoe's COM is in the gap between the prongs. The COM is a mathematical point, not necessarily a physical location.
Center of mass depends only on mass distribution. Center of gravity depends on weight distribution (mass × local gravity). They're identical in uniform gravitational fields (Earth's surface), but differ in varying gravity (e.g., a tall building experiences slightly different gravity at top vs. bottom).
Newton's Second Law applies directly: F = ma, where 'a' is the COM's acceleration. The COM moves as if all mass were concentrated there and all external forces applied at that point. Internal forces (within the system) cannot change the COM's motion—only external forces can.
Only for objects with uniform density and symmetry. A uniform sphere's COM is at its geometric center. But a hammer's COM is near the heavy head, not the midpoint of its length. If one side is denser or heavier, the COM shifts toward that side.
The pole increases rotational inertia around the wire and lowers the combined COM (walker + pole) below the wire contact point. This makes the system more stable against tipping. If they start to fall, the lowered COM creates a restoring torque that helps them regain balance—like a pendulum naturally returning to vertical.
Spacecraft rotate around their COM. By moving internal masses (fuel, reaction wheels, or astronauts), they shift the COM location, changing the rotation axis. Hubble Space Telescope uses reaction wheels positioned to control rotation about three axes through its COM. When fuel sloshes, the COM shifts unpredictably, complicating control.
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