Center of Mass
x_COM = Σmᵢxᵢ / Σmᵢ — drag 3 masses and watch the COM marker shift.
Key Notes
Center of mass (COM) is the mass-weighted average position of a system: R_COM = Σm_i·r_i / Σm_i.
For continuous bodies: R_COM = (1/M)·∫r·dm.
For symmetric uniform bodies: COM lies at the geometric center (rod, sphere, cube, ring).
COM does NOT have to lie INSIDE the body — e.g., COM of a ring or a doughnut is at the geometric center (in the hole).
Behaves as if all external force F_ext acts on a particle of total mass M at the COM: F_ext = M·a_COM.
Internal forces (between parts of the system) DO NOT affect the COM motion. Action-reaction pairs cancel.
If no external force: COM moves with constant velocity (or stays at rest).
Useful for analyzing collisions, explosions, rocket motion — all internal dynamics, no change in COM trajectory.
Formulas
Discrete COM
Mass-weighted position average.
Continuous COM
Integral form for extended bodies.
Newton's 2nd law (system)
External net force determines COM acceleration.
Velocity of COM
Total momentum / total mass.
Important Points
COM doesn't have to be inside the body — high-jumpers arch their backs so their COM passes under the bar even when their body goes over.
Internal forces (collisions, explosions, muscle forces) DO NOT change COM motion.
If F_ext = 0: COM continues with constant velocity. Useful for analyzing collisions in inertial vs COM frame.
For symmetric uniform bodies (rod, ring, sphere): COM at the geometric center.
For complicated bodies, integrate by symmetry or by decomposing into simpler parts.
Famous example: an exploding firework's COM continues on a parabolic trajectory, even as the fragments fly apart.
Center of Mass notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.