5.6 Calculating Centers of Mass and Moments of Inertia

Center of mass and moments of inertia are key concepts in physics and engineering. They help us understand how objects behave when forces are applied, especially in rotational motion.

These ideas are crucial for designing everything from buildings to spacecraft. By calculating center of mass and moments of inertia, we can predict how objects will move and balance, which is essential for many real-world applications.

Center of Mass and Moments of Inertia

Center of mass for 2D objects

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• Center of mass point where object's mass is concentrated represents average position of mass (centroid)
• For 2D object with density function $\rho(x, y)$, center of mass coordinates $(\bar{x}, \bar{y})$ given by:
  • $\bar{x} = \frac{\iint x \rho(x, y) dA}{M}$ calculates x-coordinate of center of mass
  • $\bar{y} = \frac{\iint y \rho(x, y) dA}{M}$ calculates y-coordinate of center of mass
  • $M = \iint \rho(x, y) dA$ total mass of object found by integrating density over area
• For uniform density $\rho$, formulas simplify to:
  • $\bar{x} = \frac{\iint x dA}{A}$ x-coordinate of center of mass for uniform density
  • $\bar{y} = \frac{\iint y dA}{A}$ y-coordinate of center of mass for uniform density
  • $A = \iint dA$ area of object calculated by integrating over region
• Calculating center of mass involves:
  1. Setting up double integrals for $\bar{x}$ and $\bar{y}$
  2. Determining limits of integration based on object's shape and position (rectangular, circular, triangular regions)
  3. Evaluating double integrals to find coordinates of center of mass
• Examples:
  • Center of mass of a rectangular plate lies at its geometric center
  • Center of mass of a semicircular plate lies along its axis of symmetry

Moment of inertia in planar objects

• Moment of inertia measures object's resistance to rotational acceleration about an axis
• For 2D object with density $\rho(x, y)$, moment of inertia about z-axis $I_z$ given by:
  • $I_z = \iint (x^2 + y^2) \rho(x, y) dA$ integrates product of density and squared distance from z-axis over area
• For uniform density $\rho$, formula simplifies to:
  • $I_z = \rho \iint (x^2 + y^2) dA$ density factored out of integral
• Calculating moment of inertia involves:
  1. Setting up double integral for $I_z$
  2. Determining limits of integration based on object's shape and position (rectangular, circular, triangular regions)
  3. Evaluating double integral to find moment of inertia about z-axis
• Examples:
  • Moment of inertia of a rectangular plate about an axis through its center is $\frac{1}{12}m(a^2 + b^2)$ where $m$ is mass, $a$ and $b$ are side lengths
  • Moment of inertia of a circular disk about an axis through its center is $\frac{1}{2}mr^2$ where $m$ is mass, $r$ is radius
• The radius of gyration is a measure of the distribution of mass in an object, related to its moment of inertia

Center of mass for 3D objects

• For 3D object with density $\rho(x, y, z)$, center of mass coordinates $(\bar{x}, \bar{y}, \bar{z})$ given by:
  • $\bar{x} = \frac{\iiint x \rho(x, y, z) dV}{M}$ calculates x-coordinate of center of mass
  • $\bar{y} = \frac{\iiint y \rho(x, y, z) dV}{M}$ calculates y-coordinate of center of mass
  • $\bar{z} = \frac{\iiint z \rho(x, y, z) dV}{M}$ calculates z-coordinate of center of mass
  • $M = \iiint \rho(x, y, z) dV$ total mass of object found by integrating density over volume
• For uniform density $\rho$, formulas simplify to:
  • $\bar{x} = \frac{\iiint x dV}{V}$ x-coordinate of center of mass for uniform density
  • $\bar{y} = \frac{\iiint y dV}{V}$ y-coordinate of center of mass for uniform density
  • $\bar{z} = \frac{\iiint z dV}{V}$ z-coordinate of center of mass for uniform density
  • $V = \iiint dV$ volume of object calculated by integrating over region
• Calculating center of mass involves:
  1. Setting up triple integrals for $\bar{x}$, $\bar{y}$, and $\bar{z}$
  2. Determining limits of integration based on object's shape and position (rectangular prism, cylinder, sphere)
  3. Evaluating triple integrals to find coordinates of center of mass
• Examples:
  • Center of mass of a solid cube lies at its geometric center
  • Center of mass of a solid hemisphere lies along its axis of symmetry, $\frac{3}{8}$ of the radius from the base

Advanced Concepts in Rotational Dynamics

• Torque is the rotational equivalent of force, causing angular acceleration
• Angular momentum is a measure of rotational motion, conserved in the absence of external torques
• The parallel axis theorem relates the moment of inertia about any axis to the moment of inertia about a parallel axis through the center of mass
• Principal axes of inertia are special axes for which the products of inertia vanish, simplifying rotational analysis