Moments and Centers of Mass: practice problems

Key terms · 6

center of mass

the point at which the total mass of the system could be concentrated without changing the moment

centroid

the centroid of a region is the geometric center of the region; laminas are often represented by regions in the plane; if the lamina has a constant density, the center of mass of the lamina depends only on the shape of the corresponding planar region; in this case, the center of mass of the lamina corresponds to the centroid of the representative region

lamina

a thin sheet of material; laminas are thin enough that, for mathematical purposes, they can be treated as if they are two-dimensional

moment

if n masses are arranged on a number line, the moment of the system with respect to the origin is given by $M = \sum_{i = 1}^{n} m_i x_i;$ if, instead, we consider a region in the plane, bounded above by a function $f (x)$ over an interval $[a, b],$ then the moments of the region with respect to the x- and y-axes are given by $M_x = \rho \int_a^b \frac{{[f (x)]}^2}{2} d x$ and $M_y = \rho \int_a^b x f (x) d x,$ respectively

symmetry principle

the symmetry principle states that if a region R is symmetric about a line l, then the centroid of R lies on l

theorem of Pappus for volume

this theorem states that the volume of a solid of revolution formed by revolving a region around an external axis is equal to the area of the region multiplied by the distance traveled by the centroid of the region