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Section 7.3 Center of Mass

The center of mass is the mean location of the mass of an object, and is related to the center of gravity by Newton’s Second Law because

\begin{equation*} W = mg\text{,} \end{equation*}

where \(g\) is the local strength of the gravitational field. In this course you may take \(g = \aSI{9.81}\) as a reasonable assumption for objects on the surface of the earth.

Substituting \(m_i\ g_i = W_i\) in (7.2.2) gives the equations for the center of mass.

\begin{equation} \bar{x}=\frac{\sum \bar{x}_{i} \ m_i\ g_i}{\sum \ m_i\ g_i} \quad \bar{y}=\frac{\sum \bar{y}_{i} \ m_i\ g_i}{\sum \ m_i\ g_i} \quad \bar{z}=\frac{\sum \bar{z}_{i} \ m_i\ g_i}{\sum \ m_i\ g_i}\text{.}\label{center_of_mass}\tag{7.3.1} \end{equation}

By our assumption that \(g\) is constant on the surface of the earth, \(g_i\) can be factored out of the sums and drops out of the equation completely.

\begin{equation} \bar{x}=\frac{\sum \bar{x}_{i} m_i}{\sum m_i} \quad \bar{y}=\frac{\sum \bar{y}_{i} m_i}{\sum m_i} \quad \bar{z}=\frac{\sum \bar{z}_{i} m_i}{\sum m_i}\text{.}\label{center_of_mass2}\tag{7.3.2} \end{equation}

These equations give the coordinates of the center of mass. The numerator contains the first moment of mass, and the denominator contains the total mass of the object. As long as the assumption that \(g\) is constant is valid, the center of mass and the center of gravity are identical points and the two terms may be used interchangeably.