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. and Pappus' theorem can simplify these calculations for certain shapes.
Moments and Centers of Mass
Center of mass for linear distributions
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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
xˉ=∑i=1nmi∑i=1nmixi where mi is the mass of the i-th particle and xi is its position (beads on a string)
For a continuous linear distribution, use integration to determine the center of mass
xˉ=∫dm∫x⋅dm where dm represents the differential mass element (thin rod with variable density)
Express the mass density as a function of position, ρ(x), and set up the integral xˉ=∫ρ(x)dx∫x⋅ρ(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 ρ(x,y), use double integrals to find the center of mass coordinates
xˉ=∬ρ(x,y)dA∬x⋅ρ(x,y)dA and yˉ=∬ρ(x,y)dA∬y⋅ρ(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
xˉ=∬dA∬x⋅dA and yˉ=∬dA∬y⋅dA (cardboard cutout)
Set up the double integrals for xˉ and yˉ, determine the limits of integration based on the plate's shape, and evaluate the integrals (triangular plate)
Symmetry in centroid calculations
The 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 yˉ=0 (semicircular plate)
If a plate is symmetric about the y-axis, then xˉ=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=2πyˉA where yˉ 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:
Calculate the area of the generating region using integration (semicircle)
Determine the centroid of the region using the formulas for xˉ and yˉ
Measure the distance from the centroid to the axis of revolution
Multiply the area, distance, and 2π 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
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