8.3 Center of Mass

Every object has a balance point called the center of mass (Figure 8.17). When a rigid object such as a hammer is tossed in the air, it may rotate in a complicated fashion, but its center of mass follows the same simple parabolic trajectory as a stone tossed in the air. In this section we use integration to compute the center of mass of a thin plate (also called a lamina of constant mass density \(\rho\).

The center of mass (COM) is expressed in terms of quantities called moments. The moment of a single particle of mass \(m\) with respect to a line \(L\) is the product of the particle's mass \(m\) and its directed distance (positive or negative) to the line: \[ \textrm{Moment with respect to line}~L = m \times \textrm{directed distance to}~L \]

The particular moments with respect to the \(x\)- and \(y\)-axes are denoted \(M_x\) and \(M_y\). For a particle located at the point \((x,y)\) (Figure 8.18), \[ \begin{align*} M_x = my\qquad(\textrm{mass times directed distance to} x\textrm{-axis}) \\ M_y = mx\qquad(\textrm{mass times directed distance to} y\textrm{-axis}) \end{align*} \]

By definition, moments are additive: the moment of a system of \(n\) particles with coordinates \((x_j,y_j)\) and mass \(m_j\) (Figure 8.19) is the sum \[ \begin{align*} M_x = m_1y_1 + m_2y_2 + \cdots + m_ny_n \\ M_y = m_1x_1 + m_2x_2 + \cdots + m_nx_n \end{align*} \]

The center of mass (COM) is the point \(P = (x_{\mathrm{CM}}, y_{\mathrm{CM}})\) with coordinates \[ \boxed{\bbox[#FAF8ED,5pt]{x_{\mathrm{CM}} = \frac{M_y }{M }, \qquad y_{\mathrm{CM}} = \frac{M_x}{M}}} \]

where \(M = m_1 + m_2+ \cdots + m_n\) is the total mass of the system.

Laminas (Thin Plates)

Now consider a lamina (thin plate) of constant mass density \(\rho\) occupying the region under the graph of \(f(x)\) over an interval \([a,b]\), where \(f(x)\) is continuous and \(f(x)\ge 0\) (Figure 8.21). In our calculations we will use the principle of additivity of moments mentioned above for point masses:

If a region is decomposed into smaller, non-overlapping regions, then the moment of the region is the sum of the moments of the smaller regions.

To compute the \(y\)-moment \(M_y\), we begin as usual, by dividing \([a,b]\) into \(N\) subintervals of width \(\Delta x = ({b-a})/{N}\) and endpoints \(x_j=a+j\Delta x\). This divides the lamina into \(N\) vertical strips (Figure 8.22). If \(\Delta x\) is small, the \(j\)th strip is nearly rectangular of area \(f(x_j) \Delta x\) and mass \(\rho f(x_j) \Delta x\). Since all points in the strip lie at approximately the same distance \(x_j\) from the \(y\)-axis, the moment \(M_{y,j}\) of the \(j\)th strip is approximately \[ M_{y,j}\approx (\textrm{mass}) \times (\textrm{directed distance to \(y\)-axis}) = (\rho f(x_j) \Delta x) x_j \]

By additivity of moments, \[ M_y = \sum_{j = 1}^N M_{y,j} \approx \rho\sum_{j = 1}^N x_jf(x_j) \Delta x \]

This is a Riemann sum whose value approaches \( \rho \int_a^b xf(x)\, dx\) as \(N\to \infty\), and thus \[ \boxed{\bbox[#FAF8ED,5pt]{ M_y = \rho \int_a^b x f(x)\, dx}} \]

More generally, if the lamina occupies the region between the graphs of two functions \(f_1(x)\) and \(f_2(x)\) over \([a,b]\), where \(f_1(x)\ge f_2(x)\), then \[ \boxed{\bbox[#FAF8ED,5pt]{M_y = \rho \int_a^b x (\textrm{length of vertical cut})\, dx = \rho \int_a^b x \bigl(f_1(x)-f_2(x)\bigr)\, dx}}\tag{1} \]

Think of the lamina as made up of vertical strips of length \(f_1(x)-f_2(x)\) at distance \(x\) from the \(y\)-axis (Figure 8.23).

We can compute the \(x\)-moment by dividing the lamina into horizontal strips, but this requires us to describe the lamina as a region between two curves \(x = g_1(y)\) and \(x = g_2(y)\) with \(g_1(y) \geq g_2(y)\) over an interval \([c,d]\) along the \(y\)-axis (Figure 8.24): \[ \boxed{\bbox[#FAF8ED,5pt]{M_x = \rho \int_c^d y (\textrm{length of horizontal cut})\, dy = \rho \int_c^d y \bigl(g_1(y)-g_2(y)\bigr)\, dy}}\tag{2} \]

The total mass of the lamina is \(M = \rho A\), where \(A\) is the area of the lamina: \[ \boxed{\bbox[#FAF8ED,5pt]{M = \rho A = \rho \int_a^b \bigl(f_1(x)-f_2(x)\bigr)\,dx\qquad\hbox{or}\qquad \rho\int_c^d \bigl(g_1(y)-g_2(y)\bigr)\, dy}} \]

The center-of-mass coordinates are the moments divided by the total mass: \[ \boxed{\bbox[#FAF8ED,5pt]{ x_{\mathrm{CM}} = \frac{M_y}{M}, \qquad y_{\mathrm{CM}} = \frac{M_x}{M} }} \]

A drawback of Eq. (2) for \(M_x\) is that it requires integration along the \(y\)-axis. Fortunately, there is a second formula for \(M_x\) as an integral along the \(x\)-axis. As before, divide the region into \(N\) thin vertical strips of width \(\Delta x\) (see Figure 8.26). Let \(M_{x,j}\) be the \(x\)-moment of the \(j\)th strip and let \(m_j\) be its mass. We can use the following trick to approximate \(M_{x,j}\). The strip is nearly rectangular with height \(f(x_j)\) and width \(\Delta x\), so \(m_j\approx \rho f(x_j)\,\Delta x\). Furthermore, \(M_{x,j} = y_j m_j\), where \(y_j\) is the \(y\)-coordinate of the COM of the strip. However, \(y_j\approx \frac12 f(x_j)\) because the COM of a rectangle is located at its center. Thus, \[ \begin{align*} & M_{x,j}= m_jy_j \approx\rho f(x_j)\Delta x\cdot \frac12f(x_j) = \frac12\rho f(x_j)^2\Delta x\\ & M_x =\sum_{j=1}^N M_{x,j}\approx \frac12\rho\sum_{j = 1}^N f(x_j)^2 \Delta x \end{align*} \]

This is a Riemann sum whose value approaches \({\frac{1}{2} \rho \int_a^b f(x)^2\, dx}\) as \(N\to\infty\). The case of a region between the graphs of functions \(f_1(x)\) and \(f_2(x)\) where \(f_1(x)\ge f_2(x)\ge 0\) is treated similarly, so we obtain the alternative formulas \[ \boxed{\bbox[#FAF8ED,5pt]{M_x = \frac{1}{2} \rho \int_a^b f(x)^2\, dx \qquad \textrm{or}\qquad \frac{1}{2} \rho \int_a^b \bigl(f_1(x)^2-f_2(x)^2\bigr)\, dx}}\tag{3} \]

The symmetry properties of an object give information about its centroid (Figure 8.28). For instance, the centroid of a square or circular plate is located at its center. Here is a precise formulation (see Exercise 43).

8.3.1 Summary

• The moments of a system of particles of mass \(m_j\) located at \((x_j,y_j)\) are \[ M_x = m_1y_1 + \cdots + m_ny_n,\qquad M_y = m_1x_1 + \cdots + m_nx_n \]

  The center of mass (COM) has coordinates \[ x_{\mathrm{CM}} = \dfrac{M_y}{M}\qquad\hbox{and}\qquad y_{\mathrm{CM}} = \dfrac{M_x}{M} \]

  where \(M = m_1 +\cdots + m_n\).

• Lamina (thin plate) of constant mass density \(\rho\) (region under the graph of \(f(x)\) where \(f(x)\ge 0\), or between the graphs of \(f_1(x)\) and \(f_2(x)\) where \(f_1(x)\ge f_2(x)\)): \[ M_y = \rho \int_a^b x f(x) \, dx \qquad \textrm{or}\qquad \rho \int_a^b x \bigl(f_1(x)-f_2(x)\bigr)\, dx \]
• There are two ways to compute the \(x\)-moment \(M_x\). If the lamina occupies the region between the graph of \(x = g(y)\) and the \(y\)-axis where \(g(y)\ge 0\), or between the graphs of \(g_1(y)\) and \(g_2(y)\) where \(g_1(y)\ge g_2(y)\), then \[ M_x = \rho \int_c^d y g(y) \, dy \qquad \textrm{or}\qquad \rho \int_c^d y \bigl(g_1(y)-g_2(y)\bigr)\, dy \]
• Alternative (often more convenient) formula for \(M_x\): \[ M_x = \frac{1}{2}\rho \int_a^b f(x)^2\, dx\qquad \textrm{or}\qquad \frac{1}{2} \rho \int_a^b \bigl(f_1(x)^2-f_2(x)^2\bigr)\, dx \]
• The total mass of the lamina is \( M = \rho\int_a^b \bigl(f_1(x)-f_2(x)\bigr)\, dx\). The coordinates of the center of mass (also called the centroid) are \[ x_{\mathrm{CM}} = \frac{M_y}{M},\qquad y_{\mathrm{CM}} = \frac{M_x}{M} \]
• Additivity: If a region is decomposed into smaller non-overlapping regions, then the moment of the region is the sum of the moments of the smaller regions.
• Symmetry Principle: If a lamina of constant mass density is symmetric with respect to a given line, then the center of mass (centroid) lies on that line.