In childhood you may have learned how to balance on a teeter-totter with someone significantly lighter or heavier than you. This illustrates a basic statement about the physics behind levers. In this section we will learn the mathematical generalization of this concept.
Activity67.
The purpose of this activity is to develop calculations (a model) that represent balance conditions. Note each of the images in Figure 4.5.1, Figure 4.5.2, and Figure 4.5.3 show balanced conditions.
The centroid (geometric center) of a region \(D\) is given by
\begin{equation*}
(\overline{x},\overline{y}) = \left( \dint_D x \; dA, \dint_D y \; dA \right)/\left(\dint_D 1 \; dA\right)
\end{equation*}
Note the similarity of this equation to an average. A centroid can be thought of as the average location of an region.
For many averaging problems not every point is of equal value. In these cases a weight is assigned to each point and a weighted average is calculated. When the weight is mass, then this weighted average is the center of mass.
Definition4.5.5.Center of Mass.
The center of mass of a region \(D\) is given by
\begin{equation*}
(\overline{x},\overline{y}) = \left( \dint_D x\rho(x,y) \; dA, \dint_D y\rho(x,y) \; dA \right)/\left(\dint_D \rho(x,y) \; dA\right)
\end{equation*}
ExercisesExercises
Note that these equations can be extended to higher (and lower) dimensions.
1.
Find the centroid of the triangle with vertices \((0,0),\)\((5,0)\text{,}\) and \((0,5).\)
2.
Find the center of mass of the triangle with vertices \((0,0),\)\((5,0)\text{,}\) and \((0,5)\) and density function \(\rho(x,y)=x+y.\)
3.
Find the center of mass of the region enclosed between \(y=1-x^2\) and \(y=0\) with density function \(\rho(x,y)=x^2+y.\)
4.
Find the centroid of the object enclosed within \(f(x,y)=1-x^2-y^2\) and \(z=0.\)
5.
Find the center of mass of the object enclosed within \(f(x,y)=1-x^2-y^2\) and \(z=0\) with density function \(\rho(x,y,z)=x^2+y^2.\)
6.
Find the center of mass of the region enclosed within \(r=\cos\theta\) and \(\rho(r,\theta)=r.\)
7.
Find the center of mass of the region enclosed within \(\rho=\phi\) (spherical coordinates) and density function \(d(\rho,\phi,\theta)=\rho.\)
8.
Find the center of mass of the wire structure (space curve) given by the parametric function \(x(t)=\cos t,\)\(y(t)=\sin t,\) and \(z(t)=\sin(2t)\) with density function \(\rho(x,y,z)=1.\)
