## Section 3.3 | Center of Mass

**The Finite Case**

**DEFINITION | CENTER OF MASS**

The

**center of mass**for an infinitesimally thin plate is the point, \(P\), on the plate where it will balance.

To illustrate the concepts we will be building, let's first look at the mass-less lever balanced on the fulcrum below.

Principle of the Lever from the Wolfram Demonstrations Project by S. M. Blinder

Take a moment to set \(m_1\) to 2 kg and set \(x_1\) to 3 m. Now play with different combinations of \(m_2\) and \(x_2\) to balance the lever. When you finally find a combination that yields a balance, what do you notice about the values of \(m_1\), \(m_2\), \(x_1\), and \(x_2\)?

**THEOREM | ARCHIMEDES' LAW OF THE LEVER**

Let \(d_1\) and \(d_2\) be the distances from the fulcrum to the respective masses, \(m_1\) and \(m_2\). If the fulcrum is in equilibrium, then \[d_1 m_1 = d_2 m_2.\]

We can extend this law fairly easily to the \(x\)-axis.

**THEOREM**

If masses \(m_1\) and \(m_2\) are placed on the \(x\)-axis at \(x_1\) and \(x_2\), respectively, then the center of mass of for that system of weights is at \((\overline{x},0)\), where \[\overline{x} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}.\]

**PROOF**

From the Law of the Lever, we know that \[m_1 d_1 = m_2 d_2.\]Assuming that \(x_1 \leq \overline{x} \leq x_2\), the distances from the center of mass should be \(d_1 = \overline{x} - x_1\) and \(d_2 = x_2 - \overline{x}\). Thus, \[m_1(\overline{x} - x_1) = m_2(x_2 - \overline{x})\]\[\Rightarrow (m_1 + m_2) \overline{x} = m_1 x_1 + m_2 x_2\]\[\Rightarrow \overline{x} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}\]

**DEFINITION | MOMENT**

The number \(m_i x_i\) is called the

**moment**of mass \(i\) with respect to the origin (or the \(y\)-axis if we have two-dimensional points). It represents the ability for the mass, \(m_i\), to pivot (or rotate) about the origin (\(y\)-axis).

Of course, we don't need to limit ourselves to just two masses. If we had \(n\) masses, then we can extend our theorem. Before we do, however, let's define some useful terms.

**DEFINITION | MOMENTS ABOUT THE AXES AND TOTAL MASS**

The

**moment about the \(y\)-axis**is defined to be \(M_y = \sum_{i = 1}^n{m_i x_i}\). It represents the system's ability to rotate about the \(y\)-axis. The

**moment about the \(x\)-axis**is defined to be \(M_x = \sum_{i = 1}^n{m_i y_i}\). It represents the system's ability to rotate about the \(x\)-axis. We also define the

**total mass**of the system to be \(m = \sum{m_i}\).

**THEOREM**

In general \[\overline{x} = \frac{\sum_{i = 1}^n{m_i x_i}}{\sum_{i = 1}^n{m_i}} \equiv \frac{M_y}{m}\]and \[\overline{y} = \frac{\sum_{i = 1}^n{m_i y_i}}{\sum_{i = 1}^n{m_i}} \equiv \frac{M_x}{m}.\]

You can see this theorem in action with the following Wolfram Demonstration.

Center of Mass of

*n*Points from the Wolfram Demonstrations Project by Stephen Wolfram**EXAMPLE 1 | Center of the people?**

Four people are placed on a flat, rigid plane of negligible mass itself. Abby, Bela, Cameron, and Delia have masses 62 kg, 56 kg, 78 kg, and 70 kg, respectively. Their positions with respect to the "origin" are \((-1,2)\), \((4,8)\), \((2,-1)\), and \((-3,-7)\), respectively. Where is their center of mass?

**The Infinite Case**

What happens when the system is no longer a set of masses at finite points, but instead a plate of uniform density, but possibly odd in shape? For example, the Wolfram Demonstration below (you may drag the vertices or ALT-click to add vertices).

Center of Mass of a Polygon from the Wolfram Demonstrations Project by Ken Caviness

**DEFINITION | CENTROID**

The center of mass for a lamina (a flat plate) is called the

**centroid**.

**THEOREM**

Let \[A = \int_a^b{[f(x) - g(x)] dx}.\]Then \[\overline{x} = \frac{1}{A} \int_a^b{x[f(x) - g(x)] dx}\]and\[\overline{y} = \frac{1}{A} \int_a^b{\frac{f(x)^2 - g(x)^2}{2} dx}.\]

**IMPORTANT NOTE**

The moments about the \(x\)- and \(y\)-axes are still defined as before. Specifically, \[M_y = \rho \int_a^b{x[f(x) - g(x)] dx}\]and\[M_x = \rho \int_a^b{\frac{f(x)^2 - g(x)^2}{2} dx}.\]

**EXAMPLE 2 [online #5] | Finding centroids**

Find the centroid of the region bounded by the given curves.\[y = 2 \sin{(5x)}, y = 2 \cos{(5x)}, x = 0, x = \frac{\pi}{20}\]

**EXAMPLE 3 | Finding more centroids**

Determine the centroid for the region bounded by \(y = x^4\) and \(y = x^6\) for \(0 \leq x \leq 1\).

**THEOREM | THEOREM OF PAPPUS**

Suppose we rotate a region about a line. Further suppose that this region initially lies completely on one side of this line. The volume of the resulting object is the product of the area of the region and the distance traveled by the centroid of that region during the rotation. That is, \[V = 2 \pi r_C A,\]where \(r_C\) is the distance from the centroid to the axis of rotation and \(A\) is the area of the region.

**EXAMPLE 4 [online #6] | The wonderful Theorem of Pappus**

Use the Theorem of Pappus to find the volume of a cone with height \(h\) and base radius \(r\).