6.6 Moments and Centers of Mass

Moments and centers of mass are key concepts in physics and engineering. They help us understand how objects balance and rotate. By calculating these values, we can predict an object's behavior under different forces.

Finding the center of mass is crucial for analyzing complex systems. For linear distributions and thin plates, we use integrals to determine where the mass is concentrated. Symmetry and Pappus' theorem can simplify these calculations for certain shapes.

Moments and Centers of Mass

Center of mass for linear distributions

Top images from around the web for Center of mass for linear distributions

• 

  Linear function - Wikipedia View original

  Is this image relevant?

• 

  9.6 Center of Mass | University Physics Volume 1 View original

  Is this image relevant?

• 

  multivariable calculus - Center of mass of a semi-annular plane - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Linear function - Wikipedia View original

  Is this image relevant?

• 

  9.6 Center of Mass | University Physics Volume 1 View original

  Is this image relevant?

1 of 3

Top images from around the web for Center of mass for linear distributions

• 

  Linear function - Wikipedia View original

  Is this image relevant?

• 

  9.6 Center of Mass | University Physics Volume 1 View original

  Is this image relevant?

• 

  multivariable calculus - Center of mass of a semi-annular plane - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Linear function - Wikipedia View original

  Is this image relevant?

• 

  9.6 Center of Mass | University Physics Volume 1 View original

  Is this image relevant?

1 of 3

• Represents the point where an object's mass is concentrated as if it were a single particle
• For a system of discrete particles, calculate the center of mass using the weighted average of their positions
  • $\bar{x} = \frac{\sum_{i=1}^{n} m_i x_i}{\sum_{i=1}^{n} m_i}$ where $m_i$ is the mass of the $i$-th particle and $x_i$ is its position (beads on a string)
• For a continuous linear distribution, use integration to determine the center of mass
  • $\bar{x} = \frac{\int x \cdot dm}{\int dm}$ where $dm$ represents the differential mass element (thin rod with variable density)
• Express the mass density as a function of position, $\rho(x)$, and set up the integral $\bar{x} = \frac{\int x \cdot \rho(x) dx}{\int \rho(x) dx}$
• Evaluate the integrals to find the center of mass (uniform rope suspended from one end)

Center of mass for thin plates

• For a thin plate with variable density $\rho(x, y)$, use double integrals to find the center of mass coordinates
  • $\bar{x} = \frac{\iint x \cdot \rho(x, y) dA}{\iint \rho(x, y) dA}$ and $\bar{y} = \frac{\iint y \cdot \rho(x, y) dA}{\iint \rho(x, y) dA}$ where $dA$ represents the differential area element (sheet metal with non-uniform thickness)
• For plates with uniform density, simplify the integrals by removing the density function
  • $\bar{x} = \frac{\iint x \cdot dA}{\iint dA}$ and $\bar{y} = \frac{\iint y \cdot dA}{\iint dA}$ (cardboard cutout)
• Set up the double integrals for $\bar{x}$ and $\bar{y}$, determine the limits of integration based on the plate's shape, and evaluate the integrals (triangular plate)

Symmetry in centroid calculations

• The centroid is the geometric center of a shape and coincides with the center of mass for objects with uniform density
• Utilize symmetry to simplify centroid calculations for thin plates
  • If a plate is symmetric about the x-axis, then $\bar{y} = 0$ (semicircular plate)
  • If a plate is symmetric about the y-axis, then $\bar{x} = 0$ (right triangular plate)
  • If a plate is symmetric about both axes, the centroid is at the origin (0, 0) (circular plate)
• For plates with one axis of symmetry, only one integral needs to be evaluated to find the non-zero coordinate, and the limits of integration can be simplified (half-elliptical plate)

Pappus' theorem for solids of revolution

• Relates the volume of a solid of revolution to the area and centroid of the generating shape
• For a plane region revolved about an axis in its plane, the volume is equal to the product of the area and the distance traveled by the region's centroid
  • $V = 2\pi \bar{y} A$ where $\bar{y}$ is the distance from the centroid to the axis of revolution and $A$ is the area of the region (torus generated by revolving a circle)
• To find the volume using Pappus' theorem:
  1. Calculate the area of the generating region using integration (semicircle)
  2. Determine the centroid of the region using the formulas for $\bar{x}$ and $\bar{y}$
  3. Measure the distance from the centroid to the axis of revolution
  4. Multiply the area, distance, and $2\pi$ to find the volume (solid generated by revolving a right triangle)
• Pappus' theorem can simplify volume calculations compared to the disk or shell methods, especially for complex regions (solid generated by revolving a parabolic segment)

Moments of Inertia and Rigid Bodies

• Moment of inertia measures a body's resistance to rotational acceleration (rigid body)
• Torque is the rotational equivalent of force, causing angular acceleration
• 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 the axes about which the moment of inertia is maximized or minimized