Question 5.2: The figure shown is made from a piece of thin, homogeneous w...
The figure shown is made from a piece of thin, homogeneous wire. Determine the location of its center of gravity.
STRATEGY: Since the figure is formed of homogeneous wire, its center of gravity coincides with the centroid of the corresponding line. Therefore, you can simply determine that centroid.
MODELING: Choosing the coordinate axes shown in Fig. 1 with the origin at A, determine the coordinates of the centroid of each line segment and compute the first moments with respect to the coordinate axes. You may find it convenient to list the data in a table.
| Segment | L, in. | \bar{x}, \text{ in}. | \bar{y}, \text{ in}. | \bar{x}L, \text{ in}^2 | \bar{y}L,\text{ in}^2 |
| AB | 24 | 12 | 0 | 288 | 0 |
| BC | 26 | 12 | 5 | 312 | 130 |
| CA | 10 | 0 | 5 | 0 | 50 |
| ΣL = 60 | \sum{\bar{x}}L=600 | \sum{\bar{y}}L=600 |
ANALYSIS: Substituting the values obtained from the table into the equations defining the centroid of a composite line gives
\bar{X}\sum{L}=\sum{\bar{x}L}: \quad \quad \quad \bar{X}(60 \ \text{in.})=600 \ \text{in}^2 \quad \quad \quad \quad \quad \quad \quad \bar{X}= 10 \text{ in.} \\ \bar{Y}\sum{L}=\sum{\bar{y}L}: \quad \quad \quad \bar{Y}(60 \ \text{in.})=180 \ \text{in}^2 \quad \quad \quad \quad \quad \quad \quad \bar{Y}= 3 \text{ in.}REFLECT and THINK: The centroid is not on the wire itself, but it is within the area enclosed by the wire.
