Introduction
The center of mass is the weighted average position of all mass in a system. It moves as if all external forces acted on a single particle at this point, simplifying analysis of complex systems and separating internal motions from overall translation. Understanding CM is fundamental for analyzing multi-particle systems and rigid body motion.
Definition
For discrete particles: R_CM = (Σ m_i r_i)/M where M = Σ m_i is total mass. For continuous distributions: R_CM = (∫ r dm)/M. Components: X_CM = (Σ m_i x_i)/M, Y_CM = (Σ m_i y_i)/M, Z_CM = (Σ m_i z_i)/M. The CM represents the average position weighted by mass distribution.
Properties
Center of mass may lie outside the physical body (boomerang, donut, L-shaped objects). Depends on mass distribution, not on coordinate system. For symmetric objects with uniform density, CM lies at geometric center. Useful origin choice: often convenient to place origin at CM for rotation calculations and collision analysis.
\nIntroduction
The center of mass is the weighted average position of all mass in a system. It moves as if all external forces acted on a single particle at this point, simplifying analysis of complex systems and separating internal motions from overall translation. Understanding CM is fundamental for analyzing multi-particle systems and rigid body motion.
Definition
For discrete particles: R_CM = (Σ m_i r_i)/M where M = Σ m_i is total mass. For continuous distributions: R_CM = (∫ r dm)/M. Components: X_CM = (Σ m_i x_i)/M, Y_CM = (Σ m_i y_i)/M, Z_CM = (Σ m_i z_i)/M. The CM represents the average position weighted by mass distribution.
Properties
Center of mass may lie outside the physical body (boomerang, donut, L-shaped objects). Depends on mass distribution, not on coordinate system. For symmetric objects with uniform density, CM lies at geometric center. Useful origin choice: often convenient to place origin at CM for rotation calculations and collision analysis.
\nContinuous Objects
Line mass: X_CM = (∫ x λ dx)/M where λ is linear density (mass per unit length). Area: use surface density σ = dm/dA. Volume: use volume density ρ = dm/dV. Choose coordinates to exploit symmetry - CM lies on symmetry axes and planes. For composite objects, integrate over each component.
Composite Objects
For multiple simple shapes, treat each as point mass at its own CM. Example: disk with hole - treat hole as negative mass. X_CM = (m1x1 + m2x2 + ...)/(m1 + m2 + ...) where each component's CM is known. This method simplifies calculations for complex shapes by breaking them into simpler parts.
Motion of CM
M(d2R_CM/dt2) = F_ext. Total external force determines CM acceleration. Internal forces cancel by Newton's Third Law. CM motion is independent of internal motions and forces. If F_ext = 0, CM velocity is constant regardless of complex internal dynamics. This is crucial for collision and explosion analysis.
\nSolved Example: Disk with Hole
A uniform disk of radius R = 10 cm has a hole of radius r = 4 cm drilled at distance d = 6 cm from the center. Find CM of the remaining object. Solution: Treat as complete disk (mass M) minus small disk (mass m). Let σ be surface density. M = σπR2 = σπ(100). m = σπr2 = σπ(16). Mass ratio: m/M = 16/100 = 0.16. Place origin at center of large disk. CM of complete disk is at origin. CM of small disk (the hole) is at x = d = 6 cm. CM of composite: X_CM = (M×0 - m×d)/(M - m) = -md/(M-m) = -0.16M×6/(0.84M) = -0.96/0.84 = -1.14 cm. The CM shifts 1.14 cm toward the hole (opposite side from hole center). If disk is supported at original center, it will tip toward the side with the hole.
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