Chop-Multiply-Add: Calculating Center of Mass

Section 13.2 Chop-Multiply-Add: Calculating Center of Mass

The warm-up activity below will help you explore how to calculate center of mass for a continuous object.

Exercises Warm-up Activity: The Meter Stick

You want to find the center of mass for a meter stick that has a total mass of \(0.15 \mathrm{~kg}\text{.}\)

1. Orient.

Explain why the integral below will give the length of the meter stick. Given this expression, where is the meter stick located?

\begin{equation*} \int_{0}^{1 \mathrm{~m}} dx \end{equation*}

Solution.

This integral gives the length of the board, because you are integrating over \(x\text{.}\) The result of this integral is \(1 \mathrm{~m}\text{.}\) The board is located with one end at the origin, and the other end at \(x = 1 \mathrm{~m}\text{.}\)

2. Chop.

You decide to chop up the meter stick into small pieces with infinitesimal width \(dx\text{,}\) each of which has infinitesimal mass \(dm\text{.}\) Sketch and label a diagram of the meter stick showing both \(dx\) and \(dm\text{.}\)

Solution.

Figure 13.2.1. An infinitesimal piece of a meter stick.

3. Multiply.

An infinitesimal piece of the meter stick \(dx\) and the corresponding mass of that piece \(dm\) are related by the linear mass density \(\lambda\text{:}\)

\begin{equation*} dm = \lambda dx\text{.} \end{equation*}

Calculate the density under the assumption that the density is uniform throughout the meter stick.

Solution.

The linear mass density can be written as \(\lambda=\frac{dm}{dx}\text{.}\) For a uniform mass density, this reduces to \(\lambda = M_{total} / L_{total}\text{.}\) The density only reduces to the total mass over the total length when the density is uniform. So the linear mass density is:

\begin{equation*} \lambda = \frac{M_{total}}{L_{total}} = \frac{0.15 \mathrm{~kg}}{1 \mathrm{~m}} = 0.15 \mathrm{~kg/m} \end{equation*}

4. Add.

Use an integral to determine the mass of the meter stick.

Solution.

You can determine the mass of the board by integrating both sides of \(dm = \lambda dx\text{:}\)

\begin{equation*} \int_{0}^{M} dm = \int_{0}^{L} \lambda dx \end{equation*}

\begin{equation*} M = \lambda L \end{equation*}

\begin{equation*} M = (0.15 \mathrm{~kg/m})(1 \mathrm{~m}) = 0.15 \mathrm{~kg} \end{equation*}

Watch the video below, which extends the Chop-Multiply-Add strategy above to determine a procedure for finding

Definition 13.2.2. Center of Mass for a Continuous Object.

The center of mass for a continuous system can be calculated as the average position of all parts of the object weighted by the mass at that position:

\begin{equation*} \vec{r}_{cm} = \frac{1}{M} \int \vec{r} dm \end{equation*}

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