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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 5.13 Center of Mass

Until now, you have been considering objects that are symmetrical, in which case the geometric center of the object and the center of mass are at the same position. For non-symmetrical objects, or for systems made up of multiple objects that are not arranged symmetrically, center of mass can be calculated as a weighted average.

Definition 5.13.1. Center of Mass.

The single location where you may consider an extended object to be located is known as the center of mass. It 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} \sum_{i} \vec{r}_i m_i \end{equation*}
for a set of discrete objects and
\begin{equation*} \vec{r}_{cm} = \frac{1}{M} \int \vec{r} dm \end{equation*}
for a continuous object.

Exercises Activities

A cube with side length 2L on the left next to a cube with side length L on the right, the centers of their faces touching.
Figure 5.13.2. Two adjacent cubes.

1. Calculate - Two Blocks.

Find the center of mass of the system of two cubes, assuming they are made of the same uniform material. Make sure to specify your origin! Why do you not need to integrate?

Calculate - The Board.

You want to find the center of mass of a board that has a length of 2.5 m and a total mass of 7.5 kg.
2.
Explain why the integral
\begin{equation*} \int_{0}^{2.5 \text{ m}} dx \end{equation*}
will give the length of the board. Given this expression, where is the board located?
3.
You decide to chop up the board into small pieces with infinitesimal width \(dx\text{,}\) each of which has infinitesimal mass \(dm\text{.}\) Sketch and label a diagram of the board.
4.
You know \(dx\) and \(dm\) 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 roughly uniform throughout the board.
5.
Use an integral to determine the center of mass of the board.
6.
Verify your answer by determining the mass of the board without using an integral.
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