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Learning Introductory Physics with Activities

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

Section 13.1 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 13.1.1. Center of Mass for Discrete Objects.

The center of mass for a system of discrete, pointlike objects 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*}

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 13.1.2. 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?

2. Optional: Many Blocks.

Find the \(x\)-position of the center of mass of the collection of identical, constant density blocks of mass \(M\) shown in the figure below.
Figure 13.1.3. A collection of identical blocks.
Solution.
The video below includes a discussion of center of mass and a few examples, including the solution to the Many Blocks Activity.
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