2.6 Moments and Centers of Mass

Moments and centers of mass are crucial concepts in physics and engineering. They help us understand how objects balance and rotate, whether it's a simple rod or a complex machine. These principles are essential for designing everything from bridges to spacecraft.

Calculating centers of mass involves integrating mass distributions over different shapes. For linear objects, we use line integrals. For flat objects, we use double integrals. Symmetry can simplify these calculations, making our work easier and more efficient.

Moments and Centers of Mass

Center of mass for linear distributions

• Center of mass represents the point where an object's total mass is considered concentrated
  • For a system of particles, it is the point where the weighted relative position of the distributed mass sums to zero (e.g., a collection of stars in a galaxy)
  • Formula for center of mass of a system of particles: $\bar{x} = \frac{\sum_{i=1}^{n} m_i x_i}{\sum_{i=1}^{n} m_i}$, where $m_i$ is the mass of particle $i$ and $x_i$ is its position (e.g., a molecule composed of atoms with different masses)
• Linear density $\lambda(x)$ measures mass per unit length
  • Applicable to objects with varying mass distribution along a line (e.g., a non-uniform rod)
  • Formula for center of mass of a linear distribution: $\bar{x} = \frac{\int_a^b x \lambda(x) dx}{\int_a^b \lambda(x) dx}$, where $a$ and $b$ are the endpoints of the linear distribution (e.g., a wire with varying thickness)

Center of mass for thin plates

• Thin plate is a two-dimensional object with negligible thickness
  • Surface density $\sigma(x, y)$ measures mass per unit area (e.g., a sheet of metal with non-uniform composition)
• Center of mass for a thin plate is given by $(\bar{x}, \bar{y}) = \left(\frac{\iint_R x \sigma(x, y) dA}{\iint_R \sigma(x, y) dA}, \frac{\iint_R y \sigma(x, y) dA}{\iint_R \sigma(x, y) dA}\right)$, where $R$ is the region occupied by the plate
  • For constant surface density, the formula simplifies to $(\bar{x}, \bar{y}) = \left(\frac{\iint_R x dA}{\iint_R dA}, \frac{\iint_R y dA}{\iint_R dA}\right)$ (e.g., a homogeneous rectangular plate)
• Calculating the center of mass using integration techniques involves:
  1. Evaluating the double integrals in the numerator and denominator separately (e.g., using Fubini's theorem)
  2. Converting the double integrals to iterated integrals and evaluating them using appropriate techniques like substitution or integration by parts (e.g., for a plate with a circular or triangular shape)

Symmetry in centroid location

• Centroid is the geometric center of a shape, coinciding with the center of mass for uniform density objects
• Symmetry principles simplify centroid calculations:
  • If a thin plate is symmetric about the x-axis, the y-coordinate of the centroid is 0 (e.g., a symmetric butterfly shape)
  • If a thin plate is symmetric about the y-axis, the x-coordinate of the centroid is 0 (e.g., a vertical arrow shape)
  • If a thin plate is symmetric about the origin, both the x and y-coordinates of the centroid are 0 (e.g., a perfect circle or square centered at the origin)
• Utilizing symmetry to simplify calculations involves:
  1. Identifying the axes of symmetry in the thin plate (e.g., a heart shape has vertical symmetry)
  2. Using the appropriate symmetry principle to determine one or both coordinates of the centroid (e.g., for a heart shape, x-coordinate is 0)
  3. Calculating the remaining coordinate(s) using the simplified integral(s) (e.g., only need to calculate the y-coordinate for a heart shape)

Pappus's theorem for revolution solids

• Pappus's theorem calculates the volume of a solid of revolution generated by rotating a plane figure about an axis
  • Formula for volume using Pappus's theorem: $V = 2\pi \bar{y} A$, where $\bar{y}$ is the distance from the centroid of the plane figure to the axis of rotation, and $A$ is the area of the plane figure (e.g., rotating a semicircle about its diameter)
• Steps to apply Pappus's theorem:
  1. Identify the plane figure and the axis of rotation (e.g., a right triangle rotated about one of its legs)
  2. Calculate the area of the plane figure using appropriate techniques like integration or geometric formulas (e.g., $A = \frac{1}{2}bh$ for a triangle)
  3. Determine the distance from the centroid of the plane figure to the axis of rotation using center of mass formulas or symmetry principles (e.g., for a right triangle rotated about a leg, $\bar{y} = \frac{1}{3}h$)
  4. Substitute the values for $\bar{y}$ and $A$ into the formula $V = 2\pi \bar{y} A$ to calculate the volume of the solid of revolution (e.g., for a right triangle with base $b$ and height $h$ rotated about a leg, $V = \frac{1}{3}\pi b h^2$)

Rotational dynamics and equilibrium

• Torque is the rotational equivalent of force, causing an object to rotate about an axis
• Angular momentum is a measure of rotational motion, related to an object's moment of inertia and angular velocity
• Moment of inertia represents an object's resistance to rotational acceleration, analogous to mass in linear motion
• Equilibrium occurs when the net force and net torque on a system are both zero
• A rigid body is an idealized solid object that maintains its shape under applied forces, simplifying calculations in rotational dynamics